[Paper Review] q-Euler and Genocchi numbers

Taekyun Kim|ArXiv.org|Jun 14, 2005

Advanced Mathematical Identities9 references3 citations

TL;DR

This paper introduces a new construction of q-Euler numbers different from Carlitz's, using a generating function involving q-exponential sums. From these, it defines q-Genocchi numbers and establishes their relationship to q-Bernoulli and q-Euler numbers, proving q-analogues of classical identities for sums of powers and polynomial relations.

ABSTRACT

Carlitz has introduced an interesting $q$-analogue of Frobenius-Euler numbers in [4]. He has indicated a corresponding Stadudt-Clausen theorem and also some interesting congruence properties of the $q$-Euler numbers. In this paper we give another construction of $q$-Euler numbers, which are different than his $q$-Euler numbers. By using our $q$-Euler numbers, we define the $q$-analogue of Genocchi numbers and investigate the relations between $q$-Euler numbers and $q$-analogs of Genocchi numbers.

Motivation & Objective

• To develop an alternative q-analogue of Euler numbers distinct from Carlitz's construction.
• To define q-Genocchi numbers based on the new q-Euler numbers and investigate their properties.
• To establish q-analogues of classical identities involving sums of powers and polynomial relations.
• To derive functional and recurrence relations connecting q-Euler, q-Genocchi, and q-Bernoulli numbers.

Proposed method

• Define q-Euler numbers via the generating function $ F_q(t) = [2]_q e^{t/(1-q)} \sum_{j=0}^\infty \frac{(-1)^j}{1+q^{j+1}} \left(\frac{1}{1-q}\right)^j \frac{t^j}{j!} $, yielding $ E_{n,q} $.
• Extend to q-Euler polynomials $ E_{n,q}(x) $ using a similar generating function with $ e^{xt} $, leading to a binomial-type formula.
• Define q-Genocchi numbers via $ G_q(t) = [2]_q t \sum_{n=0}^\infty (-1)^n q^n e^{[n]_q t} $, generating $ G_{n,q} $.
• Establish a relation between q-Genocchi and q-Bernoulli numbers: $ G_{n,q} = [2]_q B_{n,q} - 2[2]_q^n B_{n,q^2} $.
• Derive a q-analogue of the sum of powers identity: $ \sum_{l=0}^{n-1} (-1)^l q^l [l]_q^m = \frac{1}{[2]_q} \left( (-1)^{n+1} q^n E_{m,q}(n) - E_{m,q} \right) $.
• Prove transformation formulas for q-Genocchi polynomials under scaling, such as $ G_{n,q}(mx) = \frac{[2]_q}{[2]_{q^m}} [m]_q^{n-1} \sum_{a=0}^{m-1} (-1)^a q^{a+mx} G_{n,q^m}(x + \frac{a}{m}) $.

Experimental results

Research questions

• RQ1 How can a new q-analogue of Euler numbers be constructed that differs from Carlitz’s q-Eulerian numbers?
• RQ2 What are the defining properties and generating functions of the new q-Genocchi numbers?
• RQ3 How do the new q-Genocchi numbers relate to q-Bernoulli and q-Euler numbers?
• RQ4 What q-analogues of classical identities for sums of powers and polynomial relations can be derived from this construction?
• RQ5 Can functional equations and transformation laws be established for q-Genocchi polynomials under scaling by odd integers?

Key findings

• The new q-Euler numbers $ E_{n,q} $ are defined via a generating function involving $ q $-exponential sums and satisfy $ \lim_{q \to 1} E_{n,q} = E_n $, the classical Euler numbers.
• The q-Genocchi numbers $ G_{n,q} $ are defined through $ G_q(t) = [2]_q t \sum_{n=0}^\infty (-1)^n q^n e^{[n]_q t} $, with $ \lim_{q \to 1} G_{n,q} = G_n $, the classical Genocchi numbers.
• A key identity links q-Genocchi and q-Bernoulli numbers: $ G_{n,q} = [2]_q B_{n,q} - 2[2]_q^n B_{n,q^2} $, generalizing the classical relation $ G_n = 2nE_{n-1} $.
• A q-analogue of the sum of powers identity is established: $ \sum_{l=0}^{n-1} (-1)^l q^l [l]_q^m = \frac{1}{[2]_q} \left( (-1)^{n+1} q^n E_{m,q}(n) - E_{m,q} \right) $.
• Transformation formulas for q-Genocchi polynomials are derived, such as $ G_{n,q}(mx) = \frac{[2]_q}{[2]_{q^m}} [m]_q^{n-1} \sum_{a=0}^{m-1} (-1)^a q^{a+mx} G_{n,q^m}(x + \frac{a}{m}) $ for odd $ m $.
• The paper introduces a new operation $ * $ on functions, enabling compact expression of recurrence relations involving $ G_{n,q} $, such as $ ([m]_q - [m]_q^n)*G_{n,q} = [2]_q \sum_{k=0}^{n-1} \binom{n}{k} [m]_q^k G_{k,q^m} \sum_{a=0}^{m-1} (-1)^a q^{a(k+1)} [a]_q^{n-k} $.