1. Introduction
In 1951, Tricomi [
1] considered a sequence
of non-orthogonal polynomials related to the generalized Laguerre polynomials. These polynomials satisfy the three-term recurrence
with the initial values
and
and have ordinary generating series
They have degree
(for a generic parameter
) and can be expressed as
Tricomi polynomials have been studied from several points of view [
2,
3,
4,
5]. Here, we will show that they are related to the following tridiagonal determinants (continuants [
6,
7]). Consider the
determinants
For the first values of
n, we have
These determinants are monic polynomials of degree
n (for any parameter
) and satisfy the recurrence
with the initial values
and
. From this recurrence, it is straightforward to obtain the exponential generating series
By comparing series (
1) and (
3), we have
. For this reason, we call these polynomials Tricomi continuants. Clearly, also the Tricomi continuants can be expressed in terms of the Laguerre polynomials. Specifically, considering the Laguerre polynomials as defined in [
8], p. 108, we have the exponential generating series
and by formula (19) in [
9], Volume II, p. 189,
Then, from series (
3) and (
5), we have
or, more explicitly,
where
is the Pochhammer symbol (rising factorial).
Furthermore, from series (
3), we have that the Tricomi continuants form a Sheffer sequence. More precisely, they form the Steffensen sequence (cross-sequence) associated with the Sheffer matrix
For
, these matrices have integer entries (as can be easily deduced from formula (
6)). For instance, for
, we have the (north-west partial) matrices
Moreover, if
, we have
and then
is the Sheffer matrix of the Stirling numbers of the first kind.
In this paper, we will investigate various combinatorial and algebraic properties of the Tricomi continuants. More precisely, in
Section 2 we recall the basic definitions and properties of the Sheffer sequences and matrices. In
Section 3, we obtain the main umbral operators associated with the Tricomi continuants. Then, by using these operators, we establish some basic relations for these continuants and their derivatives. In
Section 4, we obtain a binomial identity from which we will deduce a Turan-like inequality. In
Section 5, we derive some congruences for the Tricomi continuants and their derivatives. In
Section 6, we consider the polynomials
and obtain their exponential generating series, proving that they still form a Sheffer sequence. In
Section 7, we establish a two-parameter binomial identity similar to the symmetric Carlitz identity for the Bernoulli numbers. In
Section 8, we obtain some relations between the Tricomi continuants and the Cayley continuants (another classical family of continuants also forming a Sheffer sequence). In
Section 9, we show that the infinite Hankel matrix generated by the Tricomi continuants admits an LDU-Sheffer factorization. Similarly, we show that the infinite Hankel matrix generated by the shifted Tricomi continuants admits an LTU-Sheffer factorization. Furthermore, by the first factorization, we obtain the linearization formula for the Tricomi continuants and its inverse. Finally, in
Section 10, we obtain some representations in terms of the Stirling numbers.
2. Sheffer Sequences and Sheffer Matrices
Sheffer sequences form an important class of polynomial sequences appearing in several fields of mathematics, especially in analysis, combinatorics, and umbral calculus [
8,
10,
11,
12,
13,
14,
15,
16]. In what follows, we recall the main definitions and properties we will need in the present paper. See [
17] for a historical account.
A Sheffer sequence [
18] is a polynomial sequence
having exponential generating series
where
and
are two exponential series with
,
, and
. In this case, the Sheffer sequence
has spectrum
.
An Appell sequence [
19] is a Sheffer sequence with spectrum
. A Steffensen sequence [
16] is a Sheffer sequence
, depending on a parameter
, with spectrum
, where
is an exponential series with
. Each Steffensen sequence is a cross-sequence [
13,
14], since it satisfies the binomial identity
A Sheffer matrix
is an infinite lower triangular matrix such that
for two exponential series
and
, with
,
and
. The entries of the matrix
S satisfy the recurrence
where the
are the coefficients of the series
, where
is the compositional inverse of
.
Clearly, the row polynomials of the Sheffer matrix form the Sheffer sequence with spectrum , and vice versa.
The Sheffer matrices form a group with respect to the matrix multiplication (as happens for the analogous Riordan matrices [
20,
21]). More precisely, the product of two Sheffer matrices
and
is given by
the identity matrix is
and the inverse of a Sheffer matrix
is the Sheffer matrix
Given a Sheffer matrix
and its inverse
, we have the inversion theorem: given any two sequences
and
, we have
Consider, for instance, the generalized Stirling numbers of the first kind
and the generalized Stirling numbers of the second kind
, defined as the entries of the following Sheffer matrices [
22,
23]
These matrices are related to each other by the following identities
Hence, we have the Stirling inversion theorem
or
Given a Sheffer matrix
, we have
or
, where
. A Sheffer matrix
S is a pseudo-involution when
, or, equivalently, when
is an involution. For these matrices, we have the following inversion theorem: given any two sequences
and
, we have
For instance, the binomial matrix
is a pseudo-involution. In this case, we have the binomial inversion theorem
Also the Lah matrix
whose entries are the (signless) Lah numbers, is a pseudo-involution. In this case, we have the Lah inversion theorem
Given a Sheffer matrix
, the Sheffer transform
of an exponential series
is the exponential series given by
By Pincherle’s theorem [
24], every linear operator
can be represented by means of an exponential series in the derivative
D with respect to
x. More precisely, there exists a unique polynomial sequence
, where
for every
, such that
for every polynomial
of degree
n.
Given a polynomial sequence
, where each
has degree
n, we can consider the linear operators
defined, for every
, by
The operator
J is the umbral derivative (or lowering operator, or annihilation operator), the operator
M is the umbral shift (or raising operator, or creation operator), the operator
N is the umbral theta operator, and the operator
A is the Appell operator associated with the sequence
.
The umbral operators
J,
M,
N, and
A of a Sheffer sequence
with spectrum
are given by [
8]
Some other important operators are the shift operator
defined by
and represented by
, the forward difference operator
, and the backward difference operator
. Moreover, we have
where the coefficients
are the Stirling numbers of the first kind.
5. Congruences
In this section, we will obtain some congruences for the polynomials
and
, which by formula (
6) are polynomials with integer coefficients (considering
as an arbitrary parameter). First, recall that, given two polynomials
, we have
when the corresponding coefficients of
and
are congruent modulo
p.
First of all, we have the following simple result.
Lemma 1. Let p be a prime. Then .
Proof. If
p is prime, by formula (
6), we have (working in
)
Since
, we have
. This implies our congruence. □
More generally, we have the following theorem.
Theorem 6. Let p be a prime. Then, for every , we have the congruence Proof. It is well-known that
for
p prime and
. Thus, by formula (
6), we have (working in
)
As we observed in the proof of Lemma 1, we have
. Thus, we have
, and by Lemma 1 we have congruence (
45). □
Theorem 6 can be generalized as follows.
Theorem 7. Let p be a prime. Then, for every , , we have the congruences Proof. Congruence (
46) can be proved by induction on
m, using congruence (
45). Then, congruence (
47) can be proved by induction on
s, using congruence (
46). □
Furthermore, we also have the following congruences.
Theorem 8. Let p be a prime. Then, for every , we have the congruence Proof. By formula (
36), we have (always working in
)
where we have used congruence (
6) and Wilson’s theorem
. This proves congruence (
48). □
Finally, Theorem 8 can be easily extended in the following way.
Theorem 9. Let p be a prime. Then, for every , , we have the congruences 6. A Binomial Identity
Consider the polynomials
, where
is an arbitrary parameter. To obtain their exponential generating series
, consider the bivariate generating series
Since
is the diagonal of the above bivariate series, then by Cauchy’s integral theorem (see [
27], p. 42, [
28], or [
29], p. 182), we have
There is only one pole (of the first order) tending to 0 as
, given by the unique (invertible) formal series
, such that
Hence, by the residue theorem, we have
From (
49), we have
Therefore, in conclusion, we have
where
is the Sheffer matrix
Notice also that the polynomials
form a Sheffer sequence, with spectrum
Moreover, from series (
50), we have the binomial identity
Remark 3. Using a similar approach, we can prove thatwhere is the unique formal series such that and Φ
is the Sheffer matrixAlso, in this case the polynomials form a Sheffer sequence, with spectrum 7. A Carlitz-like Identity
In 1971, Carlitz obtained [
30] the following two-parameter binomial identity
for the Bernoulli numbers
. This identity has been generalized in various ways to many other numerical and polynomial sequences. For instance, in [
31] there is a general theorem proved by the umbral calculus. Similarly, in [
32] there is a generalization to the Appell polynomials using a slightly different umbral approach. Here, we will use a similar approach to find a Carlitz-like identity for the Tricomi continuants.
First, we prove the following simple, but important, identity.
Lemma 2. The binomial transform of the Tricomi continuants is given by Proof. By following series (
3), we have
. This is equivalent to identity (
51). □
Now, we can prove the following theorem.
Theorem 10. For every , the Tricomi continuants satisfy the Carlitz-like identity Proof. Consider the umbral map defined by the linear isomorphism
where
, for all
. Then, by identity (
51) proved in Lemma 2, we have
Since
we have the umbral identity
Now, by applying
to both members of this equation, we obtain identity (
52). □
Using the same approach, we can also prove the following further identity.
Theorem 11. The Tricomi continuants satisfy the identity Proof. Consider again the umbral map
defined in the proof of Theorem 10. Since
we have the umbral identity
By applying the umbral map
to both members of this equation, we obtain identity (
53). □
Remark 4. From identity (
53)
, by the binomial inversion theorem (
11)
, we have Remark 5. For , identity (
54)
becomesThis identity can be used to obtain several other identities of the same kind. For instance, if we consider the umbral map defined by , where the are the Bell numbers [
33],
p. 493, then we have . Thus, by applying φ to both members of (
55)
, we have the identity Similarly, if we consider the harmonic numbers and , then we have the identitiesHence, by defining the umbral maps by setting and , for , and and , for , then we have and . Thus, by applying and to (
55)
, we have the identities Remark 6. The umbral method we just used to prove the previous identities has an extremely broad scope. Starting from an umbral identity (i.e., a polynomial identity), it is possible to obtain several other identities by applying a suitable umbral map [
32,
34].
Consider, for instance, Simons identity [
32,
35]
Using once again the umbral map φ defined in the proof of theorem 10
, we obtain the following Simons-like identity for the Tricomi continuants 8. Cayley Continuants
The Cayley continuants are defined by
and were considered by Cayley [
36] in relation to the Sylvester continuants (see [
6], Volume 2, pp. 425–426, [
37])
with the intention of proving the following surprising identity
where
.
Also, the Cayley continuants form a Sheffer (and a cross-) sequence [
38,
39], having exponential generating series
Moreover, they can be expressed as
where the polynomials
are the falling factorials.
Cayley and Tricomi continuants are related to each other by the following formulas
Indeed, from series (
3) and (
56), we have
and these identities are equivalent to identities (
57) and (
58), respectively.
By identity (
57), the Cayley continuants are expressed in terms of the Tricomi continuants. Now, to express the Tricomi continuants in terms of the Cayley continuants, consider the Sheffer matrix
Then, we have
or, equivalently,
Thus, replacing
x and
by
and
, we have
Therefore, by formula (
6), we have
or, equivalently,
9. Hankel Matrices
In this section, we will consider the infinite Hankel matrices
and the partial
Hankel matrices
generated by the Tricomi continuants. We will prove that the matrix
admits a Sheffer
-factorization, while the matrix
admits a Sheffer
-factorization. More precisely, an infinite matrix
A has a Sheffer
-factorization [
40] when there exist two Sheffer matrices
and
with main diagonal 1 and a diagonal matrix
D such that
. Similarly, an infinite matrix
A has a Sheffer
-factorization [
40] when there exist two Sheffer matrices
and
with main diagonal 1 and a tridiagonal matrix
T such that
.
To obtain these factorizations, we will use the exponential generating series of the infinite matrices
and
. By Taylor’s formula, they are given by
Moreover, we will use the following lemmas proved in [
40].
Lemma 3. Consider two Sheffer matricesa diagonal matrix , and the exponential generating series . Then, the exponential generating series of the matrix is given by Lemma 4. Consider two Sheffer matricesand a tridiagonal matrix , whereConsider the exponential generating series , , and . Then, the exponential generating series of the matrix is given bywhere Now, we can obtain our first factorization.
Theorem 12. The Hankel matrix admits the Sheffer LDU-factorizationwhereandMoreover, we have the Hankel determinants Proof. The exponential generating series of the Hankel matrix
can be written as
where
By Lemma 3, this implies the stated factorization. Notice that such a factorization is inherited by the partial matrices, namely . This implies the stated determinants. □
From Theorem 12, we can obtain the following identities.
Theorem 13. We have the identityand the linearization formula Proof. By Lemma 3 and Theorem 12, we have that identity (
61) is equivalent to factorization (
59). Now, we can reverse this identity by observing that in the proof of Theorem 12 we obtained the relation
or, equivalently,
Since
with
we can apply Lemma 3 where
is the Lah matrix (
12) and
. This means that we have the expansion
In conclusion, by this identity and identity (
63), we obtain the linearization formula (
62). □
In the next theorem, we obtain our second factorization.
Theorem 14. The Hankel matrix admits the Sheffer LTU-factorizationwhere is the Sheffer matrix (
60)
and , whereMoreover, we have the Hankel determinantswhere the are the polynomials defined by the exponential generating series Proof. Since
the exponential generating series of the Hankel matrix
can be written as
Since
we have
where
In particular, we have
Therefore, by Lemma 4, we have the stated factorization. Again, such a factorization is inherited by the partial matrices, namely
. Then, we have
Notice that
, where
Then, we have
where
, that is
Therefore, the polynomials
satisfy the recurrence
with the initial values
and
. Now, by the usual techniques of the formal series, it is straightforward to obtain the generating series (
64). This concludes the proof. □
Remark 7. The polynomials are themselves continuants, and by series (
64)
, they also form a Sheffer (and a cross-) sequence with spectrumAlso, these polynomials can be expressed in terms of the Laguerre polynomials. Indeed, from series (
3)
and (
4)
, we haveFor this reason, we call these polynomials Tricomi continuants of the second kind. In Section 10, we will give some representations of these continuants. Theorems 12 and 14 can be generalized as follows.
Theorem 15. The Hankel matrixadmits the Sheffer LTU-factorizationwhere is the Sheffer matrix (
60)
and is the tridiagonal matrix defined in Theorem 14. Moreover, we have the Hankel determinants Proof. By Theorems 12 and 14, we have
Now, it is straightforward to prove that
This implies the stated factorization. Then, by Theorem 14, we have the stated determinant. □
By Theorems 12 and 14, we can also compute the following Hankel determinants.
Theorem 16. The Hankel determinants of order 2 are given by Proof. Let
,
and
. Then, by Dodgson’s formula [
41] (or Jacobi identity [
42], p. 303), we have the following linear recurrence of the first order
with the initial value
. By Theorems 12 and 14, we have
and
Now, it is straightforward to show that the solution of this recurrence is given by the stated formula. □
10. Representations
In this final section, we will find some identities for the Tricomi continuants and involving the generalized Stirling numbers and, in particular, we will show that such continuants can be expressed in terms of each other.
Consider the generalized Stirling numbers defined by the Sheffer matrices (
9) and the exponential polynomials
defined by the exponential generating series
Since
we have the identity
Then, by the Stirling inversion theorem (
10), we have the representation
In particular, for
, we have the ordinary Stirling numbers of the first kind [
33], namely
and
. Therefore,
Similarly, we have the identity
and consequently
Again, by the Stirling inversion theorem (
10), we have the representation
and, in particular,
Consider now the Lah matrix (
12). Since
we have the relation
or
Finally, by the Lah inversion theorem (
13), we also have the inverse representation
These last relations are equivalent to the following matrix identities
where
and
are the Sheffer matrices (
7) and (
65).