1. Introduction
Recently, Chu [
1], using the telescoping method, obtained the following double series expressions of
and the Catalan constant
:
where
and the rising factorial
. With the works mentioned above as a source of inspiration, we derived two general double series formulas that encapsulate the Riemann zeta values
, the Catalan constant
G,
,
and several other significant mathematical constants. We highlight some identities as examples (see Equations (
20), (
22), (
24) and (
25) in
Section 3.2).
Throughout this paper, we assume that
. For complex numbers,
defined the
q-shifted factorial by [
2]
where the principal value of
is taken. For
, we introduce the notation
Jackson defined the
q-Gamma function
by
The
q-Gamma function satisfies the fundamental functional relation
The Gaussian polynomial is the
q-analogue of the binomial coefficient. It is defined by
It is known that q-analogue is a powerful tool that generalizes mathematical expressions by replacing variables with q-deformed ones. One important example of q-analogue is the Gaussian polynomial, which is a q-analogue of the binomial coefficient. The q-binomial coefficient has found applications in many areas, including q-series, combinatorics and algebraic geometry. In algebraic geometry, there exists a close relationship between q-binomial coefficients and Grassmannians.
There have been many recent developments in the study of
q-analogue. For example,
q-deformed conformal field theory has been studied in [
3,
4],
q-hypergeometric series have been studied in [
5] and
q-analogue of the Riemann zeta function has been studied in [
6,
7]. Other recent works on
q-analogue can be found in [
2,
8].
In this paper, we begin with the base of the method of telescoping sums to provide the following two main q-analogue formulas:
Theorem 1. Let be a q-function of z and be any positive integers. Then, Let
exist and the series
converge. In the above formulas, as
k and
m tend towards infinity and
q approaches
, the resulting equations can be written as follows (see Corollary 2 in
Section 2.2 and Corollary 3 in
Section 3.2).
Hence, we successfully reduce double series to single series, enabling us to quickly obtain results when computing such double series. Consequently, we can derive numerous elegant representations of classical constants in the form of double series. We list three formulas as examples (see Equations (
10) and (
11) in
Section 2.3 and Equation (
27) in
Section 3.2).
where
.
2. The First q-Formula in Theorem 1
Firstly, we utilize the q-Gamma function to create our lemma that we will employ, primarily relying on the telescoping method.
2.1. Basic Lemma
For any two complex numbers
, we define a sequence
by
This sequence clearly converges and we write the limit as
, i.e.,
It is a basic calculation that we have
Lemma 1. For any two distinct complex numbers x, y, and a positive integer k, we have Proof. By using Equations (
5) and (
6), we can establish the formula we want to prove. □
When k tends to infinity, by imposing the condition on the summation mentioned above, we arrive at the subsequent outcome.
Corollary 1. For any two complex numbers with , we have 2.2. The Proof of the First q-Formula
Moving forward, we will utilize Lemma 1 to deduce our first double
q-summation formula, Equation (
3).
We rewrite the following sum as
The inner sum can be simplified
We apply
and
in Lemma 1 and we have
Therefore, the inner sum becomes
We substitute this result into our original double sums, and then we obtain our desired result.
If we let
and
in Equation (
3), then we have
if both the series on the left and right sides of the equation converge.
Corollary 2. Let exist and the series converge. Then, for any positive integer n, we have 2.3. Examples of the First Formula
We provide some applications. Let
and
in Equation (
8). Then,
This equation appeared in [
1], Theorem 4. Based on some well-known results ([
9], Equations (15) and (19)):
We know that
. Thus, we have
We list three more identities in the following.
where
is the harmonic number. Let
in Equation (
9); this provides the identity appearing in [
1], Corollary 7:
Aside from the mentioned applications, we can also consider the case of finite summations. Simply by taking the
q parameter close to
in Equation (
3), we arrive at the following formula.
Since ([
10], Equation (5.9))
we let
and
in Equation (
12). We obtain
The following result is easily obtained from [
11], Proposition 1:
Taking
with
in Equation (
12) and
, we have
where
.
3. The Second q-Formula in Theorem 1
3.1. The Proof of the Second Formula
Let
A be the left-hand side of Equation (
4)
Rewrite
A by using Equation (
2):
By using Equation (
1), we simplify the factor
as the following
Let
be the above inner sum, that is,
A
q-analogue of Legendre duplication formula for the Gamma function [
2] is
We set
and
, respectively, and then we obtain
Substituting Equations (
16) and (
17) into
, we have
We let
,
and replace
q with
in Lemma 1; we can rewrite
as follows.
We use Equation (
15) again with
; therefore,
Substituting this result into the representation of
A, we have
Hence, we have obtained the expected equation and conclude the proof.
If we let
and
in Equation (
4), then we have
if both the series on the left and right sides of the equation converge.
Corollary 3. Let exist and the series converge. Then, for any positive integer n, we have 3.2. Examples of the Second Formula
We provide some applications. Let
and
in Equation (
19) and we obtain a symmetric double series:
Moreover, let
and
in Equation (
19), where
. Since the partial fraction decomposition is
we have
The following are the formulas with
:
Building upon the known series expansions for
G and
, we can formulate a double series expansion that specifically represents these constants. For example, we use the power seies expansion
Setting
in the above series, we have
Thus, if we let
and
in Equation (
19), then
Also, if we let
in Equation (
19), we obtain a double series representation for the Catalan constant
G:
Aliev and Dil [
12] proved that
Using Equation (
19) with
, we have
Indeed, we can derive
using the method outlined in [
13]. Moreover, by substituting
and
into Equation (
19), we obtain the following equation, which exhibits enhanced symmetry:
Lastly, we would like to emphasize that, by taking the
q parameter towards
in Equation (
4), we obtain a concise summation formula that can be applied
4. Conclusions
In this paper, our focus is on demonstrating the effective application of the "telescoping method" in handling summation expressions of
q-series (ref. Equations (
3) and (
4)). Specifically, we are primarily concerned with the summation of finite series (ref. Equations (
12), (
13) and (
28)) or infinite series (ref. Equations (
8) and (
19)) that involve coefficients represented by binomial coefficients.
Utilizing the telescoping method, we derived two general double series formulas that encompass notable mathematical constants, including the Riemann zeta values
(ref. Equations (
9), (
10) and (
21)–(
23)), the Catalan constant
G (ref. Equations (
11) and (
25)),
(ref. Equations (
20) and (
21)),
(ref. Equations (
11) and (
24)) and various other significant mathematical constants (ref. Equations (
26) and (
27)).
Interestingly, there are still many intriguing double series worth exploring, such as the work by Aliev and Dil [
12]:
or the double series for
established by Wei [
8], initially conjectured by Guo and Lian [
14]:
These equations, among others, present intriguing topics worthy of investigation. Of course, since they were derived using different approaches, it would be fascinating to obtain similar equations using the telescoping method.