The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving ₁F₂ Hypergeometric Functions That Arise from Them

Abstract

The Bessel function of the first kind $J_N\left(kx\right)$ is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind $I_N\left(kx\right)$. The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for $J_N\left(kx\right)$ useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving ${ }_1{F}_2$ hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k.

1. Introduction

Bessel functions find application in countless fields, so there has naturally been a great deal of research into how best to calculate them. Just a few of these approaches range from polynomial approximations [1,2] to solving integral equations [3] to rational approximations [4,5] to expansions in Jacobi or Chebyshev polynomials [6]. The latter are particularly prized by many workers [7,8,9].

However, the present paper has an entirely different goal from numerical approximations to Bessel functions, a representation of the Bessel function $J_N\left(kx\right)$ in a form that will be useful for analytical integration over angular variables. The work most closely related to the present approach is trigonometric approximations [10].

This need arises from the use of the Strong Field Approximation (SFA) [11,12,13,14,15,16] to calculate atomic transition amplitudes in intense laser fields. Unlike perturbation expansions that will not converge if an applied laser field is large, the SFA is an analytical approximation that is non-perturbative. Keating [17] applied it to the production of the positive antihydrogen ion and found that the reduction of the transition amplitudes to an analytic form required the expansion of the resultant Bessel functions $J_N\left(kx\right)$ in a series of spherical harmonics. I have been unable to discover such a Laplace series [18] in the literature, nor does there seem to be an expansion of the Bessel function in a series of Legendre polynomials, to which the Laplace series reduces in the common case where the function is independent of the azimuthal angle. I recreate herein Keating’s derivation, stripped of the specialized SFA terminology, and find that I can recast his coefficients from another layer of infinite series to a ${ }_2{F}_3$ generalized hypergeometric function. I also extend the method to the modified Bessel function of the first kind $I_N\left(kx\right)$.

Though designed as an analytical tool, this Fourier–Legendre series is easily converted to a series in powers; I do so as a check on its validity. Computer code is given in Appendix A for these series that provide 33-digit accuracy, matching the IEEE extended precision in some compilers.

Of further analytical significance, like-powered contributors (involving ${ }_1{F}_2$ hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying even powers of the coefficient k.

2. The Fourier–Legendre Series of a Bessel Function of the First Kind

I begin with the assumption [19] that the series

$$J_N\left(kx\right)=\sum _{L=0}^{\infty }{a}_{LN}\left(k\right)P_L\left(x\right)$$

(1)

converges uniformly. (Let D be a region in which the above series converges for each value of x. Then, the series can be said to converge uniformly in D if, for every $\epsilon >0$, there exists a number $N^{\prime }\left(\epsilon \right)$ such that, for $n>N^{\prime }$, it follows that

$$\left|J_N\left(kx\right)-\sum _{L=0}^n{a}_{LN}\left(k\right)P_L\left(x\right)\right|=\left|\sum _{L=n+1}^{\infty }{a}_{LN}\left(k\right)P_L\left(x\right)\right|<\epsilon$$

for all x in D. The coefficients are given by the orthogonality of the Legendre polynomials,

$$a_{LN}\left(k\right)=\frac{2L+1}{2}{\int }_{-1}^1{J}_N\left(kx\right)P_L\left(x\right)dx.$$

(2)

To find these coefficients, I follow Keating’s approach, first using Heine’s integral representation of the Bessel function [20] for integer indices

$$J_N\left(kx\right)=\frac{i^{-N}}{\pi }{\int }_0^{\pi }{e}^{ikxcos\theta }cos\left(N\theta \right)d\theta ,$$

(3)

so that

$$a_{LN}\left(k\right)=\frac{2L+1}{2}{\int }_{-1}^1\left[\frac{i^{-N}}{\pi }{\int }_0^{\pi }{e}^{ikxcos\theta }cos\left(N\theta \right)d\theta \right]P_L\left(x\right)dx.$$

(4)

By switching the order of integration,

$$a_{LN}\left(k\right)=\frac{2L+1}{2}\frac{i^{-N}}{\pi }{\int }_0^{\pi }\left[{\int }_{-1}^1{e}^{ikxcos\theta }{P}_L\left(x\right)dx\right]cos\left(N\theta \right)d\theta ,$$

(5)

and expanding the plane wave in a series of products of spherical Bessel functions $j_m\left(y\right)$ and Legendre polynomials [21,22],

$$e^{ikxcos\theta }=\sum _{l^{\prime }}(2l^{\prime }+1)i^{l^{\prime }}{j}_{l^{\prime }}\left(kcos\theta \right)P_{l^{\prime }}\left(x\right),$$

(6)

one has,

$$\begin{array}{ccc}{a}_{LN}\left(k\right)& =& \frac{2L+1}{2}\frac{i^{-N}}{\pi }{\int }_0^{\pi }\left[{\int }_{-1}^1\left(\sum _{l^{\prime }=0}^{\infty }(2l^{\prime }+1)i^{l^{\prime }}{j}_{l^{\prime }}\left(kcos\theta \right)P_{l^{\prime }}\left(x\right)\right)P_L\left(x\right)dx\right]cos\left(N\theta \right)d\theta \\ & =& \frac{2L+1}{2}\frac{i^{-N}}{\pi }{\int }_0^{\pi }\left[\left(\sum _{l^{\prime }=0}^{\infty }(2l^{\prime }+1)i^{l^{\prime }}{j}_{l^{\prime }}\left(kcos\theta \right)\frac{2}{2l^{\prime }+1}{\delta }_{l^{\prime }L}\right)\right]cos\left(N\theta \right)d\theta \\ & =& \left(2L+1\right)\frac{i^{L-N}}{\pi }{\int }_0^{\pi }{j}_L\left(kcos\theta \right)cos\left(N\theta \right)d\theta .\end{array}$$

(7)

Using the series expansion [23]

$$j_l\left(x\right)=\frac{1}{2}\sqrt{\pi }{\left(\frac{x}{2}\right)}^l\sum _{M=0}^{\infty }\frac{{\left(-\frac{1}{4}\right)}^M{x}^{2M}}{M!\mathsf{\Gamma }\left(l+M+\frac{3}{2}\right)}$$

(8)

this becomes

$$\begin{array}{ccc}\hfill a_{LN}\left(k\right)& =& \frac{(2L+1)}{\sqrt{\pi }}{i}^{L-\mathrm{NN}}{2}^{-L-1}\left(\sum _{M=0}^{\infty }\frac{{\left(-\frac{1}{4}\right)}^M{k}^{L+2M}}{M!\mathsf{\Gamma }\left(L+M+\frac{3}{2}\right)}\right){\int }_0^{\pi }{cos}^{L+2M}\left(\theta \right)cos\left(\mathrm{N}\theta \right)d\theta .\hfill \end{array}$$

(9)

Gröbner and Hofreiter [24] extended an integral over the interval $\left[0,\frac{\pi }{2}\right]$ that has three branches, to the interval $\left[0,\pi \right]$ with a prefactor $(1+{(-1)}^{m+n})$ that renders the central one of the three possibilities nonzero only for even values for $m+n$.

$${\int }_0^{\pi }co{s}^m\theta cos\left(n\theta \right)d\theta =\left(1+{(-1)}^{m+n}\right)\frac{\pi }{2^{m+1}}\left(\genfrac{}{}{0pt}{}{m}{\frac{m-n}{2}}\right)\left[\mathrm{where}m\ge n>-1,m-n=2K\right].$$

(10)

The other two branches, being for odd $m+n$ on $\left[0,\frac{\pi }{2}\right]$, are zero on $\left[0,\pi \right]$ when this prefactor is included. (Numerical integration also confirms that the contributions from the $\left[\frac{\pi }{2},\pi \right]$ interval cancel the contributions from the $\left[0,\frac{\pi }{2}\right]$ on these branches). The fifth edition of Gradshteyn and Ryzhik [21] (p. 417 No. 3.631.17) (in which m and n are reversed in meaning) nevertheless included all three branches and this prefactor on the interval $\left[0,\pi \right]$. By their seventh edition, Gradshteyn and Ryzhik removed this integral entirely despite the correctness of the central branch on the interval $\left[0,\pi \right]$. Neither source noted the lower limit on m that I found, that $m\ge n>-1$.

The final form for the coefficient set of the Fourier–Legendre series for the Bessel function $J_N\left(kx\right)$ is then

$$\begin{array}{ccc}{a}_{LN}\left(k\right)& =& \sqrt{\pi }(2L+1)2^{-L-1}{i}^{L-\mathrm{N}}\sum _{M=0}^{\infty }\frac{\left({\left(-\frac{1}{4}\right)}^M{k}^{L+2M}\right)}{2^{L+2M+1}\left(M!\mathsf{\Gamma }\left(L+M+\frac{3}{2}\right)\right)}\\ & \times & \left(1+{(-1)}^{L+2M+\mathrm{N}}\right)\left(\genfrac{}{}{0pt}{}{L+2M}{\frac{1}{2}(L+2M-\mathrm{N})}\right)\\ & =& \frac{\sqrt{\pi }{2}^{-2L-2}(2L+1)k^L{i}^{L-N}}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)}\left(1+{(-1)}^{L+N}\right)\left(\genfrac{}{}{0pt}{}{L}{\frac{L-N}{2}}\right)\\ & \times & { }_2{F}_3\left(\frac{L}{2}+\frac{1}{2},\frac{L}{2}+1;L+\frac{3}{2},\frac{L}{2}-\frac{N}{2}+1,\frac{L}{2}+\frac{N}{2}+1;-\frac{k^2}{4}\right)\\ & =& \sqrt{\pi }{2}^{-2L-2}(2L+1)k^L{i}^{L-N}\left(1+{(-1)}^{L+N}\right)\mathsf{\Gamma }(L+1)\\ & \times & { }_2{\tilde{F}}_3\left(\frac{L}{2}+\frac{1}{2},\frac{L}{2}+1;L+\frac{3}{2},\frac{L}{2}-\frac{N}{2}+1,\frac{L}{2}+\frac{N}{2}+1;-\frac{k^2}{4}\right)\end{array},$$

(11)

where the final two steps are new with the present work. I have included the final form using regularized hypergeometric functions [25]

$${ }_2{F}_3\left(a_1,a_2;b_1,b_2,b_3;z\right)=\mathsf{\Gamma }\left(b_1\right)\mathsf{\Gamma }\left(b_2\right)\mathsf{\Gamma }\left(b_3\right){ }_2{\tilde{F}}_3\left(a_1,a_2;b_1,b_2,b_3;z\right)$$

(12)

and cancelled the $\mathsf{\Gamma }\left(b_i\right)$ with gamma functions in the denominators of the prefactors. Each give infinities that arise whenever N > 1 is an integer larger than L and of the same parity, resulting in indeterminacies in a computation when one tries to use the conventional form of the hypergeometric function.

For the special cases of $N=0,1$, the order of the hypergeometric functions is reduced since the parameters $a_2=b_3$ and $a_1=b_2$, resp., giving

$$\begin{array}{ccc}{a}_{L0}\left(k\right)& =& \frac{\sqrt{\pi }{i}^L{2}^{-2L-2}(2L+1)k^L}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)}\left(1+{(-1)}^L\right)\left(\genfrac{}{}{0pt}{}{L}{\frac{L}{2}}\right)\\ & \times & { }_1{F}_2\left(\frac{L}{2}+\frac{1}{2};\frac{L}{2}+1,L+\frac{3}{2};-\frac{k^2}{4}\right)\\ & =& \sqrt{\pi }{i}^L{2}^{-2L-2}(2L+1)k^L\mathsf{\Gamma }\left(\frac{L}{2}+1\right)\left(1+{(-1)}^L\right)\left(\genfrac{}{}{0pt}{}{L}{\frac{L}{2}}\right)\\ & \times & { }_1{\tilde{F}}_2\left(\frac{L}{2}+\frac{1}{2};\frac{L}{2}+1,L+\frac{3}{2};-\frac{k^2}{4}\right)\end{array},$$

(13)

and

$$\begin{array}{ccc}{a}_{L1}\left(k\right)& =& \frac{\sqrt{\pi }{i}^{L-1}{2}^{-2L-2}(2L+1)k^L}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)}\left(1+{(-1)}^{L+1}\right)\left(\genfrac{}{}{0pt}{}{L}{\frac{L-1}{2}}\right)\\ & \times & { }_1{F}_2\left(\frac{L}{2}+1;\frac{L}{2}+\frac{3}{2},L+\frac{3}{2};-\frac{k^2}{4}\right)\\ & =& i^{L-1}{2}^{-L-2}(2L+1)k^L\mathsf{\Gamma }\left(\frac{L}{2}+1\right)\left(1+{(-1)}^{L+1}\right)\\ & \times & { }_1{\tilde{F}}_2\left(\frac{L}{2}+1;\frac{L}{2}+\frac{3}{2},L+\frac{3}{2};-\frac{k^2}{4}\right)\end{array},$$

(14)

In each special case, the first form involving a hypergeometric function has no indeterminacies, but I include the regularized hypergeometric function version for completeness.

3. A Numerical Check

Numerical checks are an essential element of any analytical work, particularly in checking convergence. Much of my research in physics has revolved around writing my own programs to diagonalize very large Hamiltonian matrices, and I have found that I routinely need to calculate in quadruple precision (33 digits) to gain consistent double-precision results. Since this research often contains Bessel functions, it made sense to craft a quadruple precision approximation for $J_0\left(kx\right)$ that would both allow for numerical checks and be useful for some future project of mine, or of other researchers, calculated in quadruple precision. Though Castellanos and Rosenthal obtain fifteen-decimal accuracy with their Rational Chebyshev Approximations [26] and there is a ZBESJ Bessel function double-precision package that has 18-digit accuracy (UNIVAC double precision) [27], I found no prepackaged quadruple precision routines in a search of software libraries. For this reason, I have displayed all coefficients in what follows in 33 digits, accuracy that matches the IEEE selected_real_kind (33, 4931) extended precision available on some compilers (the second argument, 4931, indicates a range of ${10}^{-4931}$ to ${10}^{4931}-1$ using 128 bits [28]). These results are given in Appendix A so that that readers may simply copy and paste them into a calculation routine.

The first 22 terms in the sum (1) are then, with $k=1$

$$\begin{array}{ccc}\hfill J_0\left(x\right)& \cong & 0.9197304100897602393144211940806200P_0\left(x\right)\hfill \\ & -& 0.1579420586258518875737139671443637P_2\left(x\right)\hfill \\ & +& 0.003438400944601109232996887872072915P_4\left(x\right)\hfill \\ & -& 0.00002919721848828729693660590986125663P_6\left(x\right)\hfill \\ & +& 1.317356952447780977655616563143280\times {10}^{-7}{P}_8\left(x\right)\hfill \\ & -& 3.684500844208203027173771096058866\times {10}^{-10}{P}_{10}\left(x\right)\hfill \\ & +& 7.011830032993845928208803328211457\times {10}^{-13}{P}_{12}\left(x\right)\hfill \\ & -& 9.665964369858912263671995372753346\times {10}^{-16}{P}_{14}\left(x\right)\hfill \\ & +& 1.009636276824546446525342170924936\times {10}^{-18}{P}_{16}\left(x\right)\hfill \\ & -& 8.266656955927637858991972584174117\times {10}^{-22}{P}_{18}\left(x\right)\hfill \\ & +& 5.448244867762758725890082837839430\times {10}^{-25}{P}_{20}\left(x\right)\hfill \\ & -& 2.952527182137354751675774606663400\times {10}^{-28}{P}_{22}\left(x\right)\hfill \\ & +& 1.338856158858534469080898670096200\times {10}^{-31}{P}_{24}\left(x\right)\hfill \\ & -& 5.154913186088512926193234837816582\times {10}^{-35}{P}_{26}\left(x\right)\hfill \\ & +& 1.706231577038503450138564028467634\times {10}^{-38}{P}_{28}\left(x\right)\hfill \\ & -& 4.906893556427796857473097979568289\times {10}^{-42}{P}_{30}\left(x\right)\hfill \\ & +& 1.237489200717479383020539576221293\times {10}^{-45}{P}_{32}\left(x\right)\hfill \\ & -& 2.759056237537871868604555688548364\times {10}^{-49}{P}_{34}\left(x\right)\hfill \\ & +& 5.477382207172712629199714648396409\times {10}^{-53}{P}_{36}\left(x\right)\hfill \\ & -& 9.744200345578852550688946057050674\times {10}^{-57}{P}_{38}\left(x\right)\hfill \\ & +& 1.562280711659504489828025148995770\times {10}^{-60}{P}_{40}\left(x\right)\hfill \\ & -& 2.269056283827394368836057470594599\times {10}^{-64}{P}_{42}\left(x\right).\hfill \end{array}$$

(15)

If one wishes to check the convergence of this series at, say, the $\left|\epsilon \right|<5\times {10}^{-8}$ level of the polynomial approximation given by E. E. Allen (and reproduced in Abramowitz and Stegun) [1],

$$\begin{array}{ccc}\hfill J_0\left(x\right)& \cong & 1-2.25{\left(\frac{x}{3}\right)}^2+1.26562{\left(\frac{x}{3}\right)}^4-0.316387{\left(\frac{x}{3}\right)}^6+0.0444479{\left(\frac{x}{3}\right)}^8\hfill \\ & -& 0.0039444{\left(\frac{x}{3}\right)}^{10}+0.0002100{\left(\frac{x}{3}\right)}^{12},\hfill \end{array}$$

(16)

at the latter’s limiting range of $x=3$ (with a value of $-0.260052$), one finds that truncating the series after the $P_{12}\left(x\right)$ term (to obtain contributions through $x^{12}$) is insufficient, giving two fewer digits of accuracy ($-0.260045$). This shows that the optimization Allen must have performed to obtain his approximation has a significant effect.

Testing accuracy at the 15-digit level shows that the series can be truncated after the $P_{24}\left(x\right)$ term in the result, $-0.260051954901933$. This $\left|\epsilon \right|<5\times {10}^{-16}$ is much better than one would expect from doubling the number of powers in the unoptimized approximation in $P_{12}\left(x\right)$. In addition, the range of applicability of the new Legendre approximation truncated after $P_{24}\left(x\right)\sim x^{24}$, if one is satisfied with $\left|\epsilon \right|<5\times {10}^{-8}$, roughly doubles to $J_0\left(6\right)\cong 0.15064531$.

Finally, in the quality checking step, I found that if a 33-digit version of (15) is imported back into the software I had been using to generate it, Mathematica 7, I only obtained a result accurate to $\left|\epsilon \right|<1\times {10}^{-32}$, but if I imported the 34-digit version actually displayed in (15), I obtained a result accurate to $\left|\epsilon \right|<1\times {10}^{-34}$ over the range $-3\le x\le 3$, with $J_0\left(3\right)\cong -0.260051954901933437624154695977331$. Accuracy exceeds $\left|\epsilon \right|<7\times {10}^{-15}$ over the range $-8\le x\le 8,$ with $J_0\left(8\right)\cong 0.171650807137560$.

4. Summing a Set of Infinite Series

If Allen’s polynomial approximation [1] has its powers of three folded into the coefficients, it becomes

$$\begin{array}{c}\hfill J_0\left(x\right)\cong 1-0.25x^2+0.0156249x^4-0.000434001x^6+6.77456\times {10}^{-6}{x}^8-6.6799\times {10}^{-8}{x}^{10}\\ \hfill +3.952\times {10}^{-10}{x}^{12}.\end{array}$$

(17)

Noting that the third term is $1/64.0004$, one is led to wonder if each of these coefficients is made up of inverse powers of primes if one were to rederive a similar approximation using a process with greater accuracy.

One can easily expand the Legendre polynomials into their constituent terms and gather like powers in (15) to give an updated polynomial approximation,

$$\begin{array}{ccc}\hfill J_0\left(x\right)& \cong & 1.000000000000000000000000000000000000x^0\hfill \\ & -& 0.2500000000000000000000000000000000000x^2\hfill \\ & +& 0.01562500000000000000000000000000000000x^4\hfill \\ & -& 0.0004340277777777777777777777777777777778x^6\hfill \\ & +& 6.781684027777777777777777777777777778\times {10}^{-6}{x}^8\hfill \\ & -& 6.781684027777777777777777777777777778\times {10}^{-8}{x}^{10}\hfill \\ & +& 4.709502797067901234567901234567901235\times {10}^{-10}{x}^{12}\hfill \\ & -& 2.402807549524439405391786344167296548\times {10}^{-12}{x}^{14}\hfill \\ & +& 9.385966990329841427311665406903502142\times {10}^{-15}{x}^{16}\hfill \\ & -& 2.896903392077111551639402903365278439\times {10}^{-17}{x}^{18}\hfill \\ & +& 7.242258480192778879098507258413196097\times {10}^{-20}{x}^{20}\hfill \\ & -& 1.496334396734045222954237036862230599\times {10}^{-22}{x}^{22}\hfill \\ & +& 2.597802772107717400962217077885817011\times {10}^{-25}{x}^{24}\hfill \\ & -& 3.842903509035084912666001594505646466\times {10}^{-28}{x}^{26}\hfill \\ & +& 4.901662639075363409012757135849038860\times {10}^{-31}{x}^{28}\hfill \\ & -& 5.446291821194848232236396817610043178\times {10}^{-34}{x}^{30}\hfill \\ & +& 5.318644356635593976793356267197307791\times {10}^{-37}{x}^{32}\hfill \\ & -& 4.600903422695150498956190542558224733\times {10}^{-40}{x}^{34}\hfill \\ & +& 3.550079801462307483762492702591222788\times {10}^{-43}{x}^{36}\hfill \\ & -& 2.458504017633176927813360597362342651\times {10}^{-46}{x}^{38}\hfill \\ & +& 1.5365650110207355798833503733514641567\times {10}^{-49}{x}^{40}\hfill \\ & -& 8.7106860035189091830121903251216788929\times {10}^{-53}{x}^{42}\hfill \\ & \cong & 1-\frac{x^2}{2^2}+\frac{x^4}{2^6}-\frac{x^6}{2^8{3}^2}+\frac{x^8}{2^{14}{3}^2}-\frac{x^{10}}{2^{16}{3}^2{5}^2}+\frac{x^{12}}{2^{20}{3}^4{5}^2}-\frac{x^{14}}{2^{22}{3}^4{5}^2{7}^2}+\frac{x^{16}}{2^{30}{3}^4{5}^2{7}^2}\hfill \\ & -& \frac{x^{18}}{2^{32}{3}^8{5}^2{7}^2}+\frac{x^{20}}{2^{36}{3}^8{5}^4{7}^2}-\frac{x^{22}}{2^{38}{3}^8{5}^4{7}^2{11}^2}+\frac{x^{24}}{2^{44}{3}^{10}{5}^4{7}^2{11}^2}-\frac{x^{26}}{2^{46}{3}^{10}{5}^4{7}^2{11}^2{13}^2}\hfill \\ & +& \frac{x^{28}}{2^{50}{3}^{10}{5}^4{7}^4{11}^2{13}^2}-\frac{x^{30}}{2^{52}{3}^{12}{5}^6{7}^4{11}^2{13}^2}+\frac{x^{32}}{2^{62}{3}^{12}{5}^6{7}^4{11}^2{13}^2}\hfill \\ & -& \frac{x^{34}}{2^{64}{3}^{12}{5}^6{7}^4{11}^2{13}^2{17}^2}+\frac{x^{36}}{2^{68}{3}^{16}{5}^6{7}^4{11}^2{13}^2{17}^2}-\frac{x^{38}}{2^{70}{3}^{16}{5}^6{7}^4{11}^2{13}^2{17}^2{19}^2}\hfill \\ & +& \frac{x^{40}}{2^{76}{3}^{16}{5}^8{7}^4{11}^2{13}^2{17}^2{19}^2}-\frac{x^{42}}{2^{78}{3}^{18}{5}^8{7}^6{11}^2{13}^2{17}^2{19}^2}.\hfill \end{array}$$

(18)

One can see from the repeating digits that the fourth through sixth lines in (18) have inverses that are powers of primes, and one can even see that the fifth term is $2^2{5}^2$ times the sixth. Subsequent terms are not at all obviously inverse powers of primes. Indeed, revealing those powers as analytic entities required many terms in excess of those required to achieve quadruple precision in the numerical results and significantly higher precision.

One might suppose that that including series terms through $P_{24}\left(x\right)$ would be sufficient for revealing the inverse powers in the coefficient of, say, the $x^{16}$ term, but that is not the case. Increasing the precision from 33 to 48 digits did not improve the situation enough. At that series truncation and with 48-digit precision, the inverse of the coefficient of the $x^{16}$ term is $\mathbf{106542032486495}.616348409991752462411671619456197,$ whose integer part is in bold face. (I used the algebra and calculus computer software Mathematica 7 for this work). This is not a product of low-level primes. Adding one more term, $P_{26}\left(x\right)$, is sufficient to bring it to $\mathbf{106542032486400}.113376300998684305400416345779209$, whose integer part is $2^{30}{3}^4{5}^2{7}^2$. One additional term makes this $\mathbf{106542032486400}.000104784167278249059923631013442$, and with every additional term added, the integer part remains the same, while the fractional part diminishes by several decimal places. The coefficient of the $x^{24}$ term required 40 terms in the series, with 48-digit precision, to establish convergence. The coefficient of the $x^{42}$ term required 74 terms in the series, with 50-digit precision, to establish convergence.

All of the coefficients in (18) include contributions from all 74 terms in the series, calculated with 50-digit precision. These were then truncated to the 37-digit precision displayed therein, except for the last two that required 38 digits. Taking their reciprocals after truncation did not forestall revealing their integer parts as powers of primes. The truncated power series gives $J_0\left(3\right)\cong -0.2600519549019334376241546959773314809$, which matches Mathematica’s BesselJ[0,3]$\cong -0.2600519549019334376241546959773314368$ (when set to 37-digit precision) within an error of $\left|\epsilon \right|<4\times {10}^{-35}$, as does the inverse prime version, $J_0\left(3\right)\cong -\frac{90658024929169559805594876257679495662633}{348615048725002045174179287542005760000000}=-0.2600519549019334376241546959773314809$.

Close examination of the inverse prime version shows that I have managed to translate into integer powers the denominators of the first 22 terms of the well-known series representation [21] (p. 970 No. 8.440)

$$J_{\nu }\left(x\right)=\sum _{k=0}^{\infty }\frac{{(-1)}^k{\left(\frac{x}{2}\right)}^{2k+\nu }}{k!\mathsf{\Gamma }(k+\nu +1)}.$$

(19)

(There is a fascinating analog to this result arising from studies of the Bessel difference equation [29]). But the outcome worth the trouble of this investigation is that this process yields a set of infinite sums whose values are inverse powers of primes. The question of why so many Legendre polynomials are required to obtain sufficiently accurate coefficients in (18) led to an examination of precisely how a given $x^n$ contributor from each Legendre polynomial contributes to the respective coefficient.

Looking back at the coefficients in (15) when multiplied by the constant terms in the Legendre polynomials that multiply them, whose first few are

$$\begin{array}{c}\hfill \left\{P_0\left(x\right)=1,P_2\left(x\right)=\frac{1}{2}\left(3x^2-1\right),P_4\left(x\right)=\frac{1}{8}\left(35x^4-30x^2+3\right),\right.\\ \hfill \left.P_6\left(x\right)=\frac{1}{16}\left(231x^6-315x^4+105x^2-5\right)\right\},\end{array}$$

(20)

there is no reason to suspect that

$$\begin{array}{c}\hfill 0.919730410089760239314421194080620\\ \hfill -\frac{1}{2}\left(-0.157942058625851887573713967144364\right)\\ \hfill +\frac{3}{8}\left(0.00343840094460110923299688787207292\right)\\ \hfill -\frac{5}{16}\left(-0.0000291972184882872969366059098612566\right)\\ \hfill +\cdots & =& 1\hfill \end{array}$$

(21)

rather than some other number close to 1, but one sees uniform convergence up through the accuracy of the calculation as one adds additional terms, as seen in

Table 1. The constant term of the Legendre series approximation as increasing numbers of terms are added from (15), to 48-digit accuracy.

0.919730410089760239314421194080619970661964806513
0.998701439402686183101278177652801821334364120020
0.999990839756911599063652010604829164640891568430
0.999999963887689188843944699951660807338623340119
0.999999999909168357337955807722426205787682516277
0.999999999999841620300892054094280728854756176645
0.99999999999999979732605041136082528291421094660
0.999999999999999999801573588581204243621535893434
0.999999999999999999999846581786952424267000275450
0.999999999999999999999999903953851991585994488442
0.999999999999999999999999999950320420042897103088
0.999999999999999999999999999999978412290228685090
0.999999999999999999999999999999999992008312509362
0.999999999999999999999999999999999999997449396440
0.999999999999999999999999999999999999999999290955
0.999999999999999999999999999999999999999999999827
1.00000000000000000000000000000000000000000000000

.

One may more formally conclude that

$$\sum _{L=0}^{\infty }{ }^{\left(2\right)}\frac{\sqrt{\pi }{i}^L(2L+1)\left(\genfrac{}{}{0pt}{}{L}{\frac{L}{2}}\right)\left(\genfrac{}{}{0pt}{}{2L}{L}\right){\left(\frac{1}{2}-\frac{L}{2}\right)}_{\frac{L}{2}}{\left(-\frac{L}{2}\right)}_{\frac{L}{2}}{2}^{-3L-2}{ }_1{F}_2\left(\frac{L}{2}+\frac{1}{2};\frac{L}{2}+1,L+\frac{3}{2};-\frac{1}{4}\right)}{\frac{L}{2}!{\left(\frac{1}{2}-L\right)}_{\frac{L}{2}}\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)}=1$$

(22)

where the superscript “(2)” on the sum indicates one is summing even values only (or one may retain the factor $\left(1+{(-1)}^L\right)$ in the sum as I do for similar equations below), which is a result I have not seen in the literature. The Pochhammer symbols ${\left(\frac{1}{2}-\frac{L}{2}\right)}_{\frac{L}{2}}$ and so on derive from a shift to [30] (p. 468 No. 7.3.1.206)

$$P_n\left(x\right)=2^{-n}\left(\genfrac{}{}{0pt}{}{2n}{n}\right)x^n{ }_2{F}_1\left(\frac{1}{2}-\frac{n}{2},-\frac{n}{2};\frac{1}{2}-n;\frac{1}{x^2}\right)$$

(23)

in the explicit sum [30] (p. 430 No. 7.2.1.1).

One may check that the series (22) converges under the Cauchy criterion. That is, if one calls a given term in (22) $u_L$ and define the partial sums

$$S_n=\sum _{L=0}^n{u}_L,$$

(24)

then the numerical series (22) is convergent if for each $\epsilon >0$ there is a number $N\left(\epsilon \right)$ such that

$$\left|S_m-S_n\right|<\epsilon$$

(25)

for all $m>n>N$. If I take, say, $\epsilon ={10}^{-21}$, it can be seen that the difference of the ninth and eighth lines $S_{10}-S_9$ in Table 1, above, fulfills this bound (with $m=10$ and $n=9$). If one instead takes $\epsilon =2\times {10}^{-46}$, it can be seen that the difference of the last two lines $S_{16}-S_{15}$ fulfills this tighter bound.

The summed series (22) was a consequence of my expansion of $J_N\left(kx\right)$ in a Fourier–Legendre series after setting $k=1$. Including k poses no problem despite its appearance as the argumemt of the ${ }_1{F}_2\left(\frac{L}{2}+\frac{1}{2};\frac{L}{2}+1,L+\frac{3}{2};-\frac{k^2}{4}\right)$ function as well as there being a $k^L$ factor in the argument of the sum. It ends up contributing a very clean factor of $k^{2h}$ to the right-hand side below. The 43 verified summed series given by the present approach (with $0\le h\le 42$) are

$$\begin{array}{ccc}\hfill \sum _{L=0}^{\infty }& & \frac{\sqrt{\pi }{i}^L{2}^{-3L-2}\left(1+{(-1)}^L\right)(2L+1)\left(\genfrac{}{}{0pt}{}{L}{\frac{L}{2}}\right)\left(\genfrac{}{}{0pt}{}{2L}{L}\right){ }_1{F}_2\left(\frac{L}{2}+\frac{1}{2};\frac{L}{2}+1,L+\frac{3}{2};-\frac{k^2}{4}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)\left(\frac{L}{2}-h\right)!}{k}^L\hfill \\ & \times & \left[\frac{{\left(\frac{1}{2}-\frac{L}{2}\right)}_{\frac{L}{2}-h}{\left(-\frac{L}{2}\right)}_{\frac{L}{2}-h}}{{\left(\frac{1}{2}-L\right)}_{\frac{L}{2}-h}}\right]=\frac{{(-1)}^h{2}^{-2h}}{h!\mathsf{\Gamma }(h+1)}{k}^{2h}.\hfill \end{array}$$

(26)

To verify the final $h=42$ summed series, I had to take the upper limit on the number of terms in the series $\ge h+74$ in order to obtain a percent difference between left- and right-hand sides that was $\le {10}^{-33}$ because the first h terms in the series do not contribute. For $h=0$, an upper limit on the number of terms in the series $\ge h+44$ was sufficient.

I wish to provide two alternative forms for readers seeking to sum a series involving ${ }_1{F}_2\left(\frac{L}{2}+\frac{1}{2};\frac{L}{2}+1,L+\frac{3}{2};-\frac{k^2}{4}\right)$ hypergeometric functions, who may have somewhat different coefficients than the above: One may use the relation [31]

$${(-a)}_n={(-1)}^n{(a-n+1)}_n$$

(27)

and the primary definition of the Pochhammer symbol [32]

$${\left(a\right)}_n=\frac{\mathsf{\Gamma }(a+n)}{\mathsf{\Gamma }\left(a\right)}(-a\in \mathbb{Z}\wedge -a\ge 0\wedge n\in \mathbb{Z}\wedge n\le -a)$$

(28)

to rewrite the term in square brackets in infinite sum (26) to give two alternative forms:

$$\begin{array}{ccc}\hfill \sum _{L=0}^{\infty }& & \frac{\sqrt{\pi }{i}^L{2}^{-3L-2}\left(1+{(-1)}^L\right)(2L+1)\left(\genfrac{}{}{0pt}{}{L}{\frac{L}{2}}\right)\left(\genfrac{}{}{0pt}{}{2L}{L}\right){ }_1{F}_2\left(\frac{L}{2}+\frac{1}{2};\frac{L}{2}+1,L+\frac{3}{2};-\frac{k^2}{4}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)\left(\frac{L}{2}-h\right)!}{k}^L\hfill \\ & \times & \left[\frac{{(-1)}^{\frac{L}{2}-h}{\left(h+\frac{1}{2}\right)}_{\frac{L}{2}-h}{(h+1)}_{\frac{L}{2}-h}}{{\left(h+\frac{L}{2}+\frac{1}{2}\right)}_{\frac{L}{2}-h}}\right]\hfill \\ & =& \sum _{L=0}^{\infty } \frac{\sqrt{\pi }{i}^L{2}^{-3L-2}\left(1+{(-1)}^L\right)(2L+1)\left(\genfrac{}{}{0pt}{}{L}{\frac{L}{2}}\right)\left(\genfrac{}{}{0pt}{}{2L}{L}\right){ }_1{F}_2\left(\frac{L}{2}+\frac{1}{2};\frac{L}{2}+1,L+\frac{3}{2};-\frac{k^2}{4}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)\left(\frac{L}{2}-h\right)!}{k}^L\hfill \\ & \times & \left[\frac{{(-1)}^{\frac{L}{2}-h}{2}^{2h-L}\mathsf{\Gamma }(L+1)\mathsf{\Gamma }\left(h+\frac{L}{2}+\frac{1}{2}\right)}{\mathsf{\Gamma }(2h+1)\mathsf{\Gamma }\left(L+\frac{1}{2}\right)}\right]=\frac{{(-1)}^h{2}^{-2h}}{h!\mathsf{\Gamma }(h+1)}{k}^{2h}.\hfill \end{array}$$

(29)

5. Series Arising from the ${\mathit{J}}_{\mathbf{1}}\left(\mathit{x}\right)$ Fourier–Legendre Series

The first 22 terms in the $J_1\left(x\right)$ Fourier–Legendre series (1) are

$$\begin{array}{ccc}\hfill J_1\left(x\right)& \cong & 0.4635981705953810635941110039338702P_1\left(x\right)\hfill \\ & -& 0.02386534565840739796307209416484866P_3\left(x\right)\hfill \\ & +& 0.0003197243559720047638524757623256028P_5\left(x\right)\hfill \\ & -& 1.970519180666594250258062929391112\times {10}^{-6}{P}_7\left(x\right)\hfill \\ & +& 6.987247473097807218791759410157014\times {10}^{-9}{P}_9\left(x\right)\hfill \\ & -& 1.610500056046875027807002442953327\times {10}^{-11}{P}_{11}\left(x\right)\hfill \\ & +& 2.607086592441628842939248193619909\times {10}^{-14}{P}_{13}\left(x\right)\hfill \\ & -& 3.127311482540796882144713619567442\times {10}^{-17}{P}_{15}\left(x\right)\hfill \\ & +& 2.891424081787050739827382596616064\times {10}^{-20}{P}_{17}\left(x\right)\hfill \\ & -& 2.123664534779369199214414455720317\times {10}^{-23}{P}_{19}\left(x\right)\hfill \\ & +& 1.269011201758673511714553707528186\times {10}^{-26}{P}_{21}\left(x\right)\hfill \\ & -& 6.290201939135925763576871358738600\times {10}^{-30}{P}_{23}\left(x\right)\hfill \\ & +& 2.628135796989325452573870774267213\times {10}^{-33}{P}_{25}\left(x\right)\hfill \\ & -& 9.381575562723076109283258050667642\times {10}^{-37}{P}_{27}\left(x\right)\hfill \\ & +& 2.894337242415984040941859061022419\times {10}^{-40}{P}_{29}\left(x\right)\hfill \\ & -& 7.794444104104171684395094261174814\times {10}^{-44}{P}_{31}\left(x\right)\hfill \\ & +& 1.848200759818170134895867052306767\times {10}^{-47}{P}_{33}\left(x\right)\hfill \\ & -& 3.888249639773912225694535890329244\times {10}^{-51}{P}_{35}\left(x\right)\hfill \\ & +& 7.306978718807123633044120058516188\times {10}^{-55}{P}_{37}\left(x\right)\hfill \\ & -& 1.234022530456621571127590099647796\times {10}^{-58}{P}_{39}\left(x\right)\hfill \\ & +& 1.883067799255568915649461884255428\times {10}^{-62}{P}_{41}\left(x\right)\hfill \\ & -& 2.609122884536350861268195351045890\times {10}^{-66}{P}_{43}\left(x\right)\hfill \end{array}$$

(30)

If one wishes to check the convergence of this series at, say, the $\left|\epsilon \right|<1.3\times {10}^{-8}$ level given by the polynomial approximation given by E. E. Allen (and reproduced in Abramowitz and Stegun) [1] (p. 370 No. 9.4.4),

$$\begin{array}{ccc}\hfill J_1\left(x\right)& \cong & \frac{x}{2}-0.062499983x^3+0.0026041448x^5-0.00005424265x^7+6.7568816\times {10}^{-7}{x}^9\hfill \\ & -& 5.3788\times {10}^{-9}{x}^{11}+2.087\times {10}^{-11}{x}^{13},\hfill \end{array}$$

(31)

at the latter’s limiting range of $x=3$ (with a value of $0.339059$), one finds that truncating the series after the $P_{13}\left(x\right)$ term (to obtain contributions through $x^{13}$) is insufficient, giving one fewer digits of accuracy ($0.339060$), again showing that the optimization Allen must have performed to obtain his approximation had a significant effect.

Testing accuracy at the 15-digit level shows that the series can be truncated after the $P_{25}\left(x\right)$ term in the result, $0.339058958525937$. This $\left|\epsilon \right|<4\times {10}^{-17}$ is better than one would expect for doubling the number of powers from the $P_{13}\left(x\right)$ truncation. In addition, the range of applicability of the new Legendre polynomial approximation truncated after $P_{24}\left(x\right)\sim x^{24}$, if one is satisfied with $\left|\epsilon \right|<1\times {10}^{-8}$, roughly doubles to $J_0\left(6\right)\cong -0.276684$.

Finally, I confirm that the series through $P_{43}\left(x\right)$ does give 33-digit accuracy for $-3\le x\le 3$, with $J_1\left(3\right)\cong 0.33905895852593645892551459720648$. Accuracy exceeds $\left|\epsilon \right|<9\times {10}^{-15}$ over the range $-8\le x\le 8,$ with $J_1\left(8\right)\cong 0.234636346853916$.

One can again expand the Legendre polynomials into their constituent terms and gather like powers in (30) to give an updated polynomial approximation,

$$\begin{array}{ccc}\hfill J_1\left(x\right)& \cong & 0.5000000000000000000000000000000000000x\hfill \\ & -& 0.06250000000000000000000000000000000000x^3\hfill \\ & +& 0.002604166666666666666666666666666666667x^5\hfill \\ & -& 0.00005425347222222222222222222222222222222x^7\hfill \\ & +& 6.781684027777777777777777777777777778\times {10}^{-7}{x}^9\hfill \\ & -& 5.651403356481481481481481481481481481\times {10}^{-9}{x}^{11}\hfill \\ & +& 3.363930569334215167548500881834215168\times {10}^{-11}{x}^{13}\hfill \\ & -& 1.501754718452774628369866465104560343\times {10}^{-13}{x}^{15}\hfill \\ & +& 5.214426105738800792950925226057501190\times {10}^{-16}{x}^{17}\hfill \\ & -& 1.448451696038555775819701451682639219\times {10}^{-18}{x}^{19}\hfill \\ & +& 3.291935672814899490499321481096907317\times {10}^{-21}{x}^{21}\hfill \\ & -& 6.234726653058521762309320986925960827\times {10}^{-24}{x}^{23}\hfill \\ & +& 9.991549123491220772931604145714680813\times {10}^{-27}{x}^{25}\hfill \\ & -& 1.372465538941101754523571998037730881\times {10}^{-29}{x}^{27}\hfill \\ & +& 1.633887546358454469670919045283012953\times {10}^{-32}{x}^{29}\hfill \\ & -& 1.701966194123390072573874005503138493\times {10}^{-35}{x}^{31}\hfill \\ & +& 1.564307163716351169645104784469796409\times {10}^{-38}{x}^{33}\hfill \\ & -& 1.278028728526430694154497372932840204\times {10}^{-41}{x}^{35}\hfill \\ & +& 9.342315267006072325690770269976902073\times {10}^{-45}{x}^{37}\hfill \\ & -& 6.146260044082942319533401493405856627\times {10}^{-48}{x}^{39}\hfill \\ & +& 3.658488121477941856865119936551105135\times {10}^{-51}{x}^{41}\hfill \\ & -& 1.979701364436115723411861437527654294\times {10}^{-54}{x}^{43}\hfill \\ & =& \frac{x}{2}-\frac{x^3}{2^4}+\frac{x^5}{2^{7}3}-\frac{x^7}{2^{11}{3}^2}+\frac{x^9}{2^{15}{3}^{2}5}-\frac{x^{11}}{2^{18}{3}^3{5}^2}+\frac{x^{13}}{2^{21}{3}^4{5}^{2}7}-\frac{x^{15}}{2^{26}{3}^4{5}^2{7}^2}\hfill \\ & +& \frac{x^{17}}{2^{31}{3}^6{5}^2{7}^2}-\frac{x^{19}}{2^{34}{3}^8{5}^3{7}^2}+\frac{x^{21}}{2^{37}{3}^8{5}^4{7}^{2}11}-\frac{x^{23}}{2^{41}{3}^9{5}^4{7}^2{11}^2}\hfill \\ & +& \frac{x^{25}}{2^{45}{3}^{10}{5}^4{7}^2{11}^{2}13}-\frac{x^{27}}{2^{48}{3}^{10}{5}^4{7}^3{11}^2{13}^2}+\frac{x^{29}}{2^{51}{3}^{11}{5}^5{7}^4{11}^2{13}^2}-\frac{x^{31}}{2^{57}{3}^{12}{5}^6{7}^4{11}^2{13}^2}\hfill \\ & +& \frac{x^{33}}{2^{63}{3}^{12}{5}^6{7}^4{11}^2{13}^{2}17}-\frac{x^{35}}{2^{66}{3}^{14}{5}^6{7}^4{11}^2{13}^2{17}^2}+\frac{x^{37}}{2^{69}{3}^{16}{5}^6{7}^4{11}^2{13}^2{17}^{2}19},\hfill \\ & -& \frac{x^{39}}{2^{73}{3}^{16}{5}^7{7}^4{11}^2{13}^2{17}^2{19}^2}+\frac{x^{41}}{2^{77}{3}^{17}{5}^8{7}^5{11}^2{13}^2{17}^2{19}^2}-\frac{x^{43}}{2^{80}{3}^{18}{5}^8{7}^6{11}^3{13}^2{17}^2{19}^2}\hfill \end{array}$$

(32)

with the latter form an integer-power realization of first 22 terms of the well-known series representation [21] (GR5 p. 970 No. 8.440). This time, all terms in the first half were truncated to the 37-digit precision displayed therein.

The truncated power series gives $J_1\left(3\right)\cong 0.339058958525936458925514597206478894$, which matches Mathematica’s BesselJ[1,3]$\cong 0.3390589585259364589255145972064788970$ (when set to 37-digit precision) within an error of $\left|\epsilon \right|<3\times {10}^{-36}$, as does the inverse prime version $J_1\left(3\right)\cong \frac{23266944578863553712347684324898325104584007}{68622120117447770997443712389847449600000000}=0.3390589585259364589255145972064788941$.

The Legendre series, via the above development, gives another set of infinite series (of which I have confirmed $0\le h\le 43$),

$$\begin{array}{ccc}\hfill \sum _{L=1}^{\infty }& & \frac{\sqrt{\pi }{i}^{L-1}\left(1+{(-1)}^{L+1}\right)(2L+1)2^{-3L-2}\left(\genfrac{}{}{0pt}{}{L}{\frac{L-1}{2}}\right)\left(\genfrac{}{}{0pt}{}{2L}{L}\right){(-1)}^{-h+\frac{L}{2}-\frac{1}{2}}}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)\left(-h+\frac{L}{2}-\frac{1}{2}\right)!}{k}^L\hfill \\ & \times & { }_1{F}_2\left(\frac{L}{2}+1;\frac{L}{2}+\frac{3}{2},L+\frac{3}{2};-\frac{k^2}{4}\right)\left[\frac{{\left(\frac{1}{2}-\frac{L}{2}\right)}_{\frac{L-1}{2}-h}{\left(-\frac{L}{2}\right)}_{\frac{L-1}{2}-h}}{{\left(\frac{1}{2}-L\right)}_{\frac{L-1}{2}-h}}\right]\hfill \\ & =& \sum _{L=1}^{\infty }{ }^{\left(2\right)}\frac{\sqrt{\pi }{i}^{L-1}\left(1+{(-1)}^{L+1}\right)(2L+1)2^{-3L-2}\left(\genfrac{}{}{0pt}{}{L}{\frac{L-1}{2}}\right)\left(\genfrac{}{}{0pt}{}{2L}{L}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)\left(-h+\frac{L}{2}-\frac{1}{2}\right)!}{k}^L\hfill \\ & \times & { }_1{F}_2\left(\frac{L}{2}+1;\frac{L}{2}+\frac{3}{2},L+\frac{3}{2};-\frac{k^2}{4}\right)\left[{(-1)}^{-h+\frac{L}{2}-\frac{1}{2}}\frac{{(h+1)}_{-h+\frac{L}{2}-\frac{1}{2}}{\left(h+\frac{3}{2}\right)}_{-h+\frac{L}{2}-\frac{1}{2}}}{{\left(h+\frac{L}{2}+1\right)}_{-h+\frac{L}{2}-\frac{1}{2}}}\right]\hfill \\ & =& \sum _{L=1}^{\infty }{ }^{\left(2\right)}\frac{\sqrt{\pi }{i}^{L-1}\left(1+{(-1)}^{L+1}\right)(2L+1)2^{-3L-2}\left(\genfrac{}{}{0pt}{}{L}{\frac{L-1}{2}}\right)\left(\genfrac{}{}{0pt}{}{2L}{L}\right)}{\mathsf{\Gamma }\left(\frac{1}{2}(2L+3)\right)\left(-h+\frac{L}{2}-\frac{1}{2}\right)!}{k}^L\hfill \\ & \times & { }_1{F}_2\left(\frac{L}{2}+1;\frac{L}{2}+\frac{3}{2},L+\frac{3}{2};-\frac{k^2}{4}\right)\left[{(-1)}^{-h+\frac{L}{2}-\frac{1}{2}}\frac{2^{2h-L+1}\mathsf{\Gamma }(L+1)\mathsf{\Gamma }\left(h+\frac{L}{2}+1\right)}{\mathsf{\Gamma }(2h+2)\mathsf{\Gamma }\left(L+\frac{1}{2}\right)}\right]\hfill \\ & =& \frac{{(-1)}^h{2}^{-2h-1}}{h!\mathsf{\Gamma }(h+2)}{k}^{2h+1}.\hfill \end{array}$$

(33)

To verify the final summed series, with $h=43$, I had to take the upper limit on the number of terms in the series $\ge h+78$ in order to obtain a percent difference between left- and right-hand sides that was $\le {10}^{-33}$ because the first h terms in the series do not contribute. For $h=0$, an upper limit on the number of terms in the series $\ge h+45$ was sufficient.

6. Series Arising from the ${\mathit{I}}_{\mathit{n}}\left(\mathit{x}\right)$ Fourier–Legendre Series

Because the modified Bessel functions of the first kind $I_N\left(kx\right)$ are related to the ordinary Bessel functions by the relation [21] (p. 961 No. 8.406.3)

$$I_n\left(z\right)=i^{-n}{J}_n\left(iz\right),$$

(34)

one merely needs to multiply by $i^{-n}$ and set $k=i$ in (11) to obtain the $I_0\left(x\right)$ Fourier–Legendre series, the first 24 terms of which are

$$\begin{array}{ccc}\hfill I_0\left(x\right)& \cong & 1.086521097023589815837941923492506P_0\left(x\right)\hfill \\ & +& 0.1758046819215242662605951354261250P_2\left(x\right)\hfill \\ & +& 0.003709009244052882533923838165527033P_4\left(x\right)\hfill \\ & +& 0.00003095105270992432198613744608777602P_6\left(x\right)\hfill \\ & +& 1.381259734719773538320052305224506\times {10}^{-7}{P}_8\left(x\right)\hfill \\ & +& 3.834312601086373005317788906125573\times {10}^{-10}{P}_{10}\left(x\right)\hfill \\ & +& 7.257172450096213936720667660411978\times {10}^{-13}{P}_{12}\left(x\right)\hfill \\ & +& 9.962746978836018020128433111635975\times {10}^{-16}{P}_{14}\left(x\right)\hfill \\ & +& 1.037251346110052630963705477046736\times {10}^{-18}{P}_{16}\left(x\right)\hfill \\ & +& 8.470496863240475339343499321604116\times {10}^{-22}{P}_{18}\left(x\right)\hfill \\ & +& 5.570541399858852219278260483523687\times {10}^{-25}{P}_{20}\left(x\right)\hfill \\ & +& 3.013347383234528850224689041823201\times {10}^{-28}{P}_{22}\left(x\right)\hfill \\ & +& 1.364338005353527272638479093175249\times {10}^{-31}{P}_{24}\left(x\right)\hfill \\ & +& 5.246088467162281648944660989359565\times {10}^{-35}{P}_{26}\left(x\right)\hfill \\ & +& 1.734417052236546525979562610169336\times {10}^{-38}{P}_{28}\left(x\right)\hfill \\ & +& 4.982929889631203560686967762401821\times {10}^{-42}{P}_{30}\left(x\right)\hfill \\ & +& 1.255546430559877621587201790700357\times {10}^{-45}{P}_{32}\left(x\right)\hfill \\ & +& 2.797096596401706413444068821508193\times {10}^{-49}{P}_{34}\left(x\right)\hfill \\ & +& 5.548955677049963483909673845071489\times {10}^{-53}{P}_{36}\left(x\right)\hfill \\ & +& 9.865206225205083247212985096531573\times {10}^{-57}{P}_{38}\left(x\right)\hfill \\ & +& 1.580763691652306983099443761944673\times {10}^{-60}{P}_{40}\left(x\right)\hfill \\ & +& 2.294688331479205281600814719914093\times {10}^{-64}{P}_{42}\left(x\right)\hfill \\ & +& 3.031771495580703895127109933386607\times {10}^{-68}{P}_{44}\left(x\right)\hfill \\ & +& 3.661200772680598752990852186025167\times {10}^{-72}{P}_{46}\left(x\right).\hfill \end{array}$$

(35)

If one wishes to check the convergence of this series at, say, the $\left|\epsilon \right|<1.6\times {10}^{-7}$ level of the polynomial approximation given by E. E. Allen (and reproduced in Abramowitz and Stegun) [1] (p. 378 No. 9.8.1),

$$\begin{array}{ccc}\hfill I_0\left(x\right)& \cong & 1+3.5156229{\left(\frac{x}{3.75}\right)}^2+3.0899424{\left(\frac{x}{3.75}\right)}^4+1.2067492{\left(\frac{x}{3.75}\right)}^6\hfill \\ & +& 0.2659732{\left(\frac{x}{3.75}\right)}^8+0.0360768{\left(\frac{x}{3.75}\right)}^{10}+0.0045813{\left(\frac{x}{3.75}\right)}^{12}\hfill \\ & =& 1+0.25x^2+0.0156252x^4+0.00043394x^6+6.801234\times {10}^{-6}{x}^8+6.56017\times {10}^{-8}{x}^{10}\hfill \\ & +& 5.9240\times {10}^{-10}{x}^{12}\hfill \end{array}$$

(36)

at the latter’s limiting range of $x=3.75$ (with a value of $9.11895$), one finds that truncating the series after the $P_{12}\left(x\right)$ term (to obtain contributions through $x^{12}$) is insufficient, giving two fewer digits of accuracy ($9.1187$). This again shows that the optimization Allen must have performed to obtain his approximation has a significant effect.

Testing accuracy at the 15-digit level shows that the series can be truncated after the $P_{28}\left(x\right)$ term in the result, $9.1189458608445666$, with $\left|\epsilon \right|<5\times {10}^{-17}$. In addition, the range of applicability of the new Legendre approximation truncated after $P_{24}\left(x\right)\sim x^{24}$, if one is satisfied with $\left|\epsilon \right|<1.6\times {10}^{-7}$, roughly doubles to $I_0\left(7.5\right)\cong 268.1613$.

Finally, in the quality checking step, I found that if I imported a 33-digit version of (35) back into the software I had been using to generate it, Mathematica 7, I only obtained a result accurate to $\left|\epsilon \right|<1\times {10}^{-32}$, but if I imported the 34-digit version version actually displayed in (35), I obtained a result accurate to $\left|\epsilon \right|<1\times {10}^{-34}$ over the range $-3.75\le x\le 3.75$, with $I_0\left(3.75\right)\cong 9.118945860844566690670997606599715$. Accuracy exceeds $\left|\epsilon \right|<6\times {10}^{-14}$ over the range $-8\le x\le 8,$ with $I_0\left(8\right)\cong 427.5641157218$.

If one turns one’s attention to the latter form of the polynomial approximation (36) and compare it with the $J_0$ polynomial approximation (17), one sees that the $I_0$ version is approximately the $J_0$ version with all of the negative signs reversed. That the correspondence is not exact for the higher-power terms is likely a result of the optimization scheme in the two cases having slightly different ranges of validity.

One may apply (34) to (19), giving $i^{\nu }{(-1)}^k{i}^{2k+\nu }{\left(\frac{x}{2}\right)}^{2k+\nu }={(-1)}^{\nu }{\left(\frac{x}{2}\right)}^{2k+\nu }$ for each numerator, which indeed gives the series [21] (GR5 p. 971 No. 8.445)

$$I_{\nu }\left(x\right)=\sum _{k=0}^{\infty }\frac{{\left(\frac{x}{2}\right)}^{2k+\nu }}{k!\mathsf{\Gamma }(k+\nu +1)}.$$

(37)

I, thus, need not display the 24-term polynomial approximation for $I_0$ apart from the two terms unneeded for accuracy in the $J_0$ version that are necessary for $I_0$:

$$\begin{array}{ccc}\hfill ...& +& 4.4993212828093539168451396307446688497\times {10}^{-56}{x}^{44}\hfill \\ & +& 2.1263333094562164068266255343783879252\times {10}^{-59}{x}^{46}.\hfill \end{array}$$

(38)

One obtains the other terms by simply negating the negative signs in (18). This is not true of (35) because the arguments of the Legendre polynomials do not undergo $x\to ix$ since they derive from the definition of the Fourier–Legendre series, (1). The k-dependence is entirely within the coefficients $a_{LN}\left(k\right)$.

Furthermore, the $I_0$ Legendre series expansion leads to no new set of summed series since these would simply be (26) with $k=i\kappa$.

Although (1) allows one to easily compute the Fourier–Legendre series for any $J_n\left(x\right)$ or $I_n\left(x\right)$, to enable readers to find these series for higher indices by recursion [21] (p. 979 No. 8.471.1, p. 981 No. 8.486.1) I give the first 24 terms in the $I_1\left(x\right)$ Fourier–Legendre series to complete the required pair:

$$\begin{array}{ccc}\hfill I_1\left(x\right)& \cong & 0.5386343421852555592809081051666336P_1\left(x\right)\hfill \\ & +& 0.02618069164825977449795296407260333P_3\left(x\right)\hfill \\ & +& 0.0003419851912550806236210094361507344P_5\left(x\right)\hfill \\ & +& 2.077651971699656963860267070724864\times {10}^{-6}{P}_7\left(x\right)\hfill \\ & +& 7.299001518662431414905576324932877\times {10}^{-9}{P}_9\left(x\right)\hfill \\ & +& 1.671443482954853739162527767203215\times {10}^{-11}{P}_{11}\left(x\right)\hfill \\ & +& 2.692744551459235232734936666452704\times {10}^{-14}{P}_{13}\left(x\right)\hfill \\ & +& 3.218106754771162455853759838545282\times {10}^{-17}{P}_{15}\left(x\right)\hfill \\ & +& 2.966624646773403074824196435937542\times {10}^{-20}{P}_{17}\left(x\right)\hfill \\ & +& 2.173686883720901031568436047655748\times {10}^{-23}{P}_{19}\left(x\right)\hfill \\ & +& 1.296326265789554875546711294998344\times {10}^{-26}{P}_{21}\left(x\right)\hfill \\ & +& 6.414855102151415733596588296578833\times {10}^{-30}{P}_{23}\left(x\right)\hfill \\ & +& 2.676389925142875786863074285387196\times {10}^{-33}{P}_{25}\left(x\right)\hfill \\ & +& 9.542035089444710700263714658817980\times {10}^{-37}{P}_{27}\left(x\right)\hfill \\ & +& 2.940669572337884276201779460203862\times {10}^{-40}{P}_{29}\left(x\right)\hfill \\ & +& 7.911705033029330504663434638574930\times {10}^{-44}{P}_{31}\left(x\right)\hfill \\ & +& 1.874426565726980007813411585706840\times {10}^{-47}{P}_{33}\left(x\right)\hfill \\ & +& 3.940459072597980181771454632976250\times {10}^{-51}{P}_{35}\left(x\right)\hfill \\ & +& 7.400090413796917559009186360838868\times {10}^{-55}{P}_{37}\left(x\right)\hfill \\ & +& 1.248984620737396858084740490332061\times {10}^{-58}{P}_{39}\left(x\right)\hfill \\ & +& 1.904842982553207494042180613785837\times {10}^{-62}{P}_{41}\left(x\right)\hfill \\ & +& 2.637959760920312924684635466402215\times {10}^{-66}{P}_{43}\left(x\right)\hfill \\ & +& 3.332061910821697596383220274010501\times {10}^{-70}{P}_{45}\left(x\right).\hfill \end{array}$$

(39)

If one wishes to check the convergence of this series at, say, the $\left|\epsilon \right|<8\times {10}^{-9}$ level given by the polynomial approximation given by E. E. Allen (and reproduced in Abramowitz and Stegun) [1] (p. 378 No. 9.8.3),

$$\begin{array}{ccc}\hfill I_1\left(x\right)& \cong & \frac{x}{2}+0.0625x^3+0.00260419x^5+0.0000542445x^7+6.79868\times {10}^{-7}{x}^9+5.48303\times {10}^{-9}{x}^{11}\hfill \\ & +& 4.191\times {10}^{-11}{x}^{13},\hfill \end{array}$$

(40)

at the latter’s limiting range of $x=3.75$ (with a value of $7.78002$), one finds that truncating the series after the $P_{13}\left(x\right)$ term (to obtain contributions through $x^{13}$) is insufficient, giving three fewer digits of accuracy ($7.77996$). Yet again this shows that the optimization Allen must have performed to obtain his approximation has a significant effect.

Testing accuracy at the 15-digit level shows that the series can be truncated after the $P_{27}\left(x\right)$ term in the result, $7.780015229824415$, with $\left|\epsilon \right|<4\times {10}^{-16}$. In addition, the range of applicability of the new Legendre approximation truncated after $P_{27}\left(x\right)\sim x^{27}$, if one is satisfied with $\left|\epsilon \right|<5\times {10}^{-9}$, roughly doubles to $I_1\left(6.5\right)\cong 97.735011$.

Finally, in my quality checking step, I found that if I imported a 33-digit version of (39) back into the software I had been using to generate it, Mathematica 7, I only obtained a result accurate to $\left|\epsilon \right|<1\times {10}^{-32}$, but if I imported the 34-digit version actually displayed in (35), I obtained a result accurate to $\left|\epsilon \right|<1\times {10}^{-34}$ over the range $-3.75\le x\le 3.75$, with $I_1\left(3.75\right)\cong 7.780015229824415864988676277516113$. Accuracy exceeds $\left|\epsilon \right|<1\times {10}^{-18}$ over the range $-8\le x\le 8,$ with $I_1\left(8\right)\cong 399.873136782560098$.

Because the above-noted correspondence between the power-series versions of $J_0\left(x\right)$ and $I_0\left(x\right)$ (reversing all of the negative signs in the former to achieve the latter) applies as well to $J_1\left(x\right)$ and $I_1\left(x\right)$, there is no need to display the power-series version of the latter except for the additional term,

$$...+9.781133223498595471402477458140584456\times {10}^{-58}{x}^{45}.$$

(41)

7. Conclusions

I have found the Fourier–Legendre series of modified Bessel functions of the first kind $I_N\left(kx\right)$ based on that found by Keating [17] for the Bessel functions of the first kind $J_N\left(kx\right)$ and have shown that Keating’s coefficients, comprised of infinite series, can be reduced to ${ }_2{F}_3$ functions. For $N=0$ and 1, I gave numerical values for those coefficients up through $x^{46}$ with 33-digit accuracy.

Each of these infinite Fourier–Legendre series may be decomposed into an infinite sum of infinite series, by gathering like powers from the Legendre polynomials in each of the terms in the Fourier–Legendre series. I showed that each of these infinite sub-series converges to values that are inverse powers of the first eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k. Given the relative paucity of infinite series whose values are known (e.g., 24 pages in Gradshteyn and Ryzhik compared to their 900 pages of known integrals), having even one such to add to the total has the potential to be of use to future researchers. Herein I have added an infinite set of infinite series of ${ }_1{F}_2$ functions whose values are now known.