[Paper Review] On the q-Extensions of the Bernoulli and Euler Numbers, Related Identities and Lerch Zeta Function

Taekyun Kim, Young‐Hee Kim+1 more|ArXiv.org|Jan 2, 2009

Advanced Mathematical Identities19 references8 citations

TL;DR

This paper introduces q-extensions of λ-Bernoulli and λ-Euler numbers using p-adic q-integrals, deriving new identities and establishing connections to the Lerch zeta function. The key contribution is the construction of q-analogues of the Lerch zeta function via generating functions of λ-q-Bernoulli and λ-q-Euler numbers, with the result that ζ_q(λ,1−k) = −β_k,q(λ)/k for k ∈ ℕ.

ABSTRACT

Recently, $λ$-Bernoulli and $λ$-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the $q$-extensions of the $λ$-Bernoulli and the $λ$-Euler numbers by using the bosonic $p$-adic $q$-integral and the fermionic $p$-adic $q$-integral. The investigation of these $λ$-$q$-Bernoulli and $λ$-$q$-Euler numbers leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning $q$-Bernoulli and $q$-Euler numbers. By using derivative operator to the generating functions of $λ$-$q$-Bernoulli and $λ$-$q$-Euler numbers, we give the $q$-extensions of Lerch zeta function.

Motivation & Objective

• To systematically develop q-extensions of λ-Bernoulli and λ-Euler numbers using p-adic q-integrals.
• To generalize earlier results on q-Bernoulli and q-Euler numbers by incorporating λ-deformations.
• To establish new identities involving λ-q-Bernoulli and λ-q-Euler numbers through integral representations.
• To define and analyze q-extensions of the Lerch zeta function using generating functions of λ-q-special numbers.
• To unify and extend previous results on zeta functions and special number sequences in the q- and λ-deformed settings.

Proposed method

• Utilizes the bosonic p-adic q-integral I_q(f) = ∫_{ℤ_p} f(x) dμ_q(x) and the fermionic p-adic q-integral I_{-q}(f) = ∫_{ℤ_p} f(x) dμ_{-q}(x) to define λ-q-Bernoulli and λ-q-Euler numbers.
• Applies the derivative operator to generating functions of λ-q-Bernoulli and λ-q-Euler numbers to derive q-extensions of the Lerch zeta function.
• Employs the generating function g*(t,λ:q) = ∫_{ℤ_p} λ^x e^{[x]_q t} dμ_{-q}(x) = [2]_q ∑_{m=0}^∞ (−1)^m λ^m q^m e^{[m]_q t} for the DC-type λ-q-Euler polynomials.
• Derives recurrence relations such as q^n I_{-q}(f_n) = (−1)^n I_{-q}(f) + [2]_q ∑_{l=0}^{n-1} (−1)^{n−1−l} q^l f(l) for p-adic q-integrals.
• Constructs two q-extensions of the Lerch zeta function: ζ_q(λ,s) and ζ_q^*(λ,s), based on q-series involving [m]_q^s.
• Uses the identity β_k,q(λ)/k = −∑_{m=1}^∞ q^m λ^m [m]_q^{k−1} to link the q-Bernoulli numbers to the zeta function at negative integers.

Experimental results

Research questions

• RQ1 How can λ-Bernoulli and λ-Euler numbers be generalized to q-analogues using p-adic q-integrals?
• RQ2 What identities emerge from the q-extensions of λ-Bernoulli and λ-Euler numbers?
• RQ3 How do the generating functions of λ-q-Bernoulli and λ-q-Euler numbers lead to q-extensions of the Lerch zeta function?
• RQ4 What is the functional relationship between the q-extended Lerch zeta function and the λ-q-Bernoulli numbers?
• RQ5 Can the q-extensions of the Lerch zeta function be expressed in terms of q-series and special number sequences?

Key findings

• The q-extended Lerch zeta function ζ_q(λ,s) is defined as (1−q)(2−s)/(s−1) ∑_{m=1}^∞ q^m λ^m / [m]_q^{s−1} + ∑_{m=1}^∞ q^m λ^m / [m]_q^s for |q|<1 and λ = e^{2πi/f}.
• The functional equation ζ_q(λ,1−k) = −β_k,q(λ)/k holds for all k ∈ ℕ, linking the q-extended zeta function to the λ-q-Bernoulli numbers.
• A second q-extension ζ_q^*(λ,s) is defined as ∑_{m=1}^∞ q^m λ^m / [m]_q^s, which satisfies ζ_q^*(λ,1−k) = −β_k,q(λ)/k.
• The DC-type λ-q-Euler numbers are defined via E_n,q^*(λ) = ∫_{ℤ_p} λ^x [x]_q^n dμ_{-q}(x) = [2]_q ∑_{m=0}^∞ (−1)^m λ^m q^m [m]_q^n.
• The generating function for DC-type λ-q-Euler polynomials is ∑_{n=0}^∞ E_n,q^*(λ,x) t^n / n! = [2]_q ∑_{m=0}^∞ (−1)^m λ^m q^m e^{[m+x]_q t}.
• The results generalize earlier works on q-Bernoulli and q-Euler numbers, and unify them with λ-deformations and p-adic integral methods.