On Multiple Generalized 𝑤-Genocchi Polynomials and Their Applications

Abstract

We define the multiple generalized 𝑤-Genocchi polynomials. By using fermionic 𝑝-adic invariant integrals, we derive some identities on these generalized 𝑤-Genocchi polynomials, for example, fermionic 𝑝-adic integral representation, Witt's type formula, explicit formula, multiplication formula, and recurrence formula for these 𝑤-Genocchi polynomials.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, ℤ, ℤ_𝑝, ℚ_𝑝, ℂ, and ℂ_𝑝 will, respectively, denote the ring of integers, the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, the complex number field, and. the 𝑝-adic completion of the algebraic closure of ℚ_𝑝. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝜈𝑝(𝑝)}=𝑝⁻¹.

The 𝑞-basic natural numbers are defined by [𝑛]_𝑞=1−𝑞^𝑛1−𝑞=1+𝑞+𝑞²+⋯+𝑞^𝑛−1 (1.1) for 𝑛∈ℕ, and the binomial coefficient is defined as 𝑛𝑘=𝑛!()=𝑛()⋯()𝑛−𝑘!𝑘!𝑛−1𝑛−𝑘+1.𝑘! (1.2) The binomial formulas are well known that (1−𝑏)^𝑛=^𝑛_𝑖=0𝑛𝑖(−1)^𝑖𝑏^𝑖,1(1−𝑏)^𝑛=^𝑛_𝑖=0𝑖𝑏𝑛+𝑖−1_𝑖 (1.3)(see, [1, 2]). When one talks of 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex number 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ, one normally assumes that |𝑞|<1. If 𝑞∈ℂ_𝑝, one normally assumes that |𝑞−1|_𝑝<1. We use the notation [𝑥]_𝑞=1−𝑞^𝑥,[𝑥]1−𝑞^−𝑞=1−−𝑞^𝑥,1+𝑞 (1.4) see [1–13] for all 𝑥∈ℤ_𝑝. Note that lim_𝑞→1[𝑥]_𝑞=𝑥 for 𝑥∈ℤ_𝑝 in presented 𝑝-adic case.

Let 𝑈𝐷(ℤ_𝑝) be denoted by the set of uniformly differentiable function on ℤ_𝑝. For 𝑓∈𝑈𝐷(ℤ_𝑝), an invariant 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined as 𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑛→∞1+𝑞1+𝑞^{𝑝𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥)−𝑞^𝑥. (1.5) Thus, we have the following integral relation: lim_𝑞→1𝑞𝐼_−𝑞𝑓₁+𝐼_−𝑞𝑓=1+𝑞𝑓(0), (1.6) where 𝑓₁(𝑥)=𝑓(𝑥+1), and the fermionic 𝑝-adic invariant integral relation: 𝐼₋₁𝑓=lim_𝑞→1𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥),𝐼₋₁𝑓₁+𝐼𝑓=2𝑓(0). (1.7)(1.8)

Now, we recall that the definitions of 𝑤-Euler polynomials and 𝑤-Genocchi polynomials are defined as 2𝑒^𝑥𝑡𝑤𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛,𝑤(𝑥)𝑡^𝑛,𝑛!2𝑡𝑒^𝑥𝑡𝑤𝑒^𝑡=+1^∞_𝑛=0𝐺_𝑛,𝑤(𝑥)𝑡^𝑛𝑛!,𝑡∈ℝ,𝑤∈ℂ, (1.9) with |1−𝑤|_𝑝<1, respectively. In the special case 𝑥=0, 𝐸_𝑛,𝑤(0)=𝐸_𝑛,𝑤, and 𝐺_𝑛,𝑤(0)=𝐺_𝑛,𝑤 are called 𝑤-Euler numbers and 𝑤-Genocchi numbers (see [2, 9]).

In [13], Bayard and Simsek have studied multiple generalized Bernoulli polynomials as follows: ^𝑟_𝑗=1𝑎_𝑗𝑡+log𝑤^𝑎𝑗(𝑤𝑒^𝑡)^𝑎𝑗𝑒−1^𝑡=^∞_𝑛=0𝐵^(𝑟)_𝑛,𝑤𝑥;𝑎₁,…𝑎_𝑟𝑡^𝑛,𝑤𝜋𝑛!𝑡+log(||)<min𝑎₁⋯𝜋𝑎_𝑝, (1.10) where 𝑎₁,…,𝑎_𝑟 are strictly positive real numbers.

The purpose of this paper is to define another construction of multiple generalized 𝑤-Genocchi polynomials and numbers, which are different from multiple generalized Bernoulli polynomials and numbers in [13]. By using fermionic 𝑝-adic invariant integrals, we derive some identities on these generalized 𝑤-Genocchi polynomials, for example, fermionic 𝑝-adic integral representation, Witt's type formula, explicit formulas, multiplication formula, and recurrence formula for these 𝑤-Genocchi polynomials.

2. Multiple Generalized 𝑤-Genocchi Polynomials and Numbers

Let 𝑟∈ℕ and 𝑎₁,…,𝑎_𝑟 be strictly positive real numbers. The multiple generalized 𝑤-Genocchi polynomials 𝐺^(𝑟)_𝑛,𝑤(𝑥;𝑎₁,…,𝑎_𝑟) are defined as ^𝑟_𝑗=1(2𝑡)^𝑟(𝑤𝑒^𝑡)^𝑎𝑗𝑒+1^𝑥𝑡=^∞_𝑛=0𝐺^(𝑟)_𝑛,𝑤𝑥;𝑎₁,…𝑎_𝑟𝑡^𝑛,𝑛!for𝑡∈ℝ,𝑤∈ℂ, (2.1) where |log𝑤+𝑡|≤min_{1≤𝑗≤𝑟}{𝜋/𝑎_𝑗}. The values of 𝐺^(𝑟)_𝑛,𝑤(𝑥;𝑎₁,…,𝑎_𝑟) at 𝑥=0 are called the multiple generalized 𝑤-Genocchi numbers: when 𝑟=1, 𝑤=1, and 𝑎_𝑗=0 (𝑗=1,…𝑟), the polynomials or numbers are called the ordinary Genocchi polynomials or numbers.

It is known that 𝑡_ℤ𝑝𝑤^𝑧𝑒^{𝑡(𝑧+𝑥)}𝑑𝜇₋₁(𝑧)=2𝑡𝑤𝑒^𝑡=+1^∞_𝑛=0𝐺_𝑛,𝑤(𝑥)𝑡^𝑛,𝑡𝑛!_𝑟ℤ𝑝⋯_ℤ𝑝𝑤^{𝑧1+𝑧2+⋯+𝑧𝑟}𝑒^{𝑡(𝑧1+⋯+𝑧𝑟+𝑥)}𝑑𝜇₋₁𝑧₁⋯𝑑𝜇₋₁𝑧_𝑟=2𝑡𝑤𝑒^𝑡+1^𝑟𝑒^𝑥𝑡=^∞_𝑛=0𝐺^(𝑟)_𝑛,𝑤(𝑥)𝑡^𝑛.𝑛! (2.2)

In fact, let us take 𝑡∈ℝ,𝑤∈ℂ, and we apply the above difference integral formula (1.8) for 𝑓(𝑧)=𝑤^𝑎𝑧𝑒^𝑡𝑎𝑧, then we obtain 2(𝑤𝑒^𝑡)^𝑎𝑒+1^𝑡𝑥=_ℤ𝑝𝑤^𝑎𝑧𝑒^{𝑡(𝑎𝑧+𝑥)}𝑑𝜇₋₁(𝑧). (2.3) By (2.3), we easily see that ^𝑟_𝑗=1(2𝑡)^𝑟(𝑤𝑒^𝑡)^𝑎𝑗𝑒+1^𝑥𝑡=𝑡^𝑟_ℤ𝑝⋯_ℤ𝑝𝑤^{𝑎1𝑧1+⋯+𝑎𝑟𝑧𝑟}𝑒^{𝑡(𝑎1𝑧1+⋯+𝑎𝑟𝑧𝑟+𝑥)}𝑑𝜇₋₁𝑧₁⋯𝑑𝜇₋₁𝑧_𝑟=^∞_{𝑛=0ℤ𝑝}⋯_ℤ𝑝𝑤^{𝑎1𝑧1+⋯+𝑎𝑟𝑧𝑟}𝑎₁𝑧₁+⋯+𝑎_𝑟𝑧_𝑟+𝑥^𝑛×𝑑𝜇₋₁𝑧₁⋯𝑑𝜇₋₁𝑧_𝑟𝑡^𝑛+𝑟,𝑛!𝐺^(𝑟)_0,𝑤𝑥;𝑎₁,…,𝑎_𝑟=⋯=𝐺^(𝑟)_{𝑟−1,𝑤}𝑥;𝑎₁,…,𝑎_𝑟=0. (2.4)(2.5)

By (2.4) and (2.5), we obtain the following fermionic 𝑝-adic integral representation formula for these numbers.

Theorem 2.1 (𝑝-adic integral representation). Let 𝑟∈ℕ and 𝑎₁,…,𝑎_𝑟 be strictly positive real numbers. Then one has a fermionic 𝑝-adic invariant integral representation for the multiple generalized 𝑤-Genocchi polynomials 𝐺^(𝑟)_𝑛,𝑤(𝑥;𝑎₁,…,𝑎_𝑟) as follows: 𝐺^(𝑟)_{𝑛+𝑟,𝑤}𝑥;𝑎₁,…,𝑎_𝑟𝑟!(_𝑟^𝑛+𝑟)=_ℤ𝑝⋯_ℤ𝑝𝑤^{𝑎1𝑧1+⋯+𝑎𝑟𝑧𝑟}𝑎₁𝑧₁+⋯+𝑎_𝑟𝑧_𝑟+𝑥^𝑛𝑑𝜇₋₁𝑧₁⋯𝑑𝜇₋₁𝑧_𝑟 (2.6) for 𝑛≥𝑟 and 𝐺^(𝑟)_0,𝑤𝑥;𝑎₁,…,𝑎_𝑟=⋯=𝐺^(𝑟)_{𝑟−1,𝑤}𝑥;𝑎₁…,𝑎_𝑟=0. (2.7)

We remark that if we set 𝑟=1 and 𝑎₁=1, then we have the following equation: 𝐺⁽¹⁾_{𝑛+𝑟,𝑤}(𝑥;1)1!₁^𝑛+1=𝐺^(𝑟)_𝑛+1,𝑤(𝑥)𝑛+1=𝐸_𝑛,𝑤(𝑥). (2.8) The generalized 𝑤-Genocchi polynomials are given by 2𝑡(𝑤𝑒^𝑡)^𝑎𝑒+1^𝑥𝑡=^∞_𝑛=0𝐺_𝑛,𝑤()𝑡𝑥;𝑎^𝑛,𝑛!_ℤ𝑝𝑤^𝑎𝑧𝑒^{𝑡(𝑎𝑧+𝑥)}𝑑𝜇₋₁(𝑧)=^∞_{𝑛=0ℤ𝑝}𝑤^𝑎𝑧(𝑎𝑧+𝑥)^𝑛𝑑𝜇₋₁(𝑧)𝑡^𝑛. (2.9) By comparing the coefficients on both sides in (2.9), we obtain the following identity on the generalized 𝑤-Genocchi polynomials 𝐺_𝑛,𝑤(𝑥;𝑎)𝑛!=_ℤ𝑝𝑤^𝑎𝑧(𝑎𝑧+𝑥)^𝑛𝑑𝜇₋₁(𝑧). (2.10) Similarly, from (2.4), we can obtain the following Witt's type formula for the multiple generalized 𝑤-Genocchi polynomials.

Theorem 2.2 (Witt's type formula). Let 𝑟∈ℕ and 𝑎₁,…,𝑎_𝑟 be strictly positive real numbers. Then one has 𝐺^(𝑟)_𝑛,𝑤𝑥;𝑎₁,…,𝑎_𝑟=𝑛!_ℤ𝑝⋯_ℤ𝑝𝑤^{𝑎1𝑧1+⋯+𝑎𝑟𝑧𝑟}𝑎₁𝑧₁+⋯+𝑎_𝑟𝑧_𝑟+𝑥^𝑛𝑑𝜇₋₁𝑧₁⋯𝑑𝜇₋₁𝑧_𝑟. (2.11)

From (2.4), we can directly calculate the following: 𝐺^(𝑟)_𝑛,𝑤𝑥;𝑎₁,…,𝑎_𝑟=_ℤ𝑝⋯_ℤ𝑝𝑤^{𝑎1𝑧1+⋯+𝑎𝑟𝑧𝑟}𝑎₁𝑧₁+⋯+𝑎_𝑟𝑧_𝑟+𝑥^𝑛×𝑑𝜇₋₁𝑧₁⋯𝑑𝜇₋₁𝑧_𝑟=𝑛!_𝑛𝑖=0𝑛𝑖𝑥^𝑛−𝑖_ℤ𝑝⋯_ℤ𝑝𝑤^{𝑎1𝑧1+⋯+𝑎𝑟𝑧𝑟}×𝑎₁𝑧₁+⋯+𝑎_𝑟𝑧_𝑟^𝑖𝑑𝜇₋₁𝑧₁⋯𝑑𝜇₋₁𝑧_𝑟=𝑛!_𝑛𝑖=0𝑛𝑖²(𝑛−𝑖)!𝑥^𝑛−𝑖𝐺^(𝑟)_𝑖,𝑤𝑎₁,…,𝑎_𝑟. (2.12) From (2.12), we get the following explicit formula.

Theorem 2.3 (explicit formula). Let 𝑟∈ℕ and 𝑎₁,…,𝑎_𝑟 be strictly positive real numbers. Then one has 𝐺^(𝑟)_𝑛,𝑤𝑥;𝑎₁,…,𝑎_𝑟=^𝑛_𝑖=0𝑛𝑖²(𝑛−𝑖)!𝑥^𝑛−𝑖𝐺^(𝑟)_𝑖,𝑤𝑎₁,…,𝑎_𝑟. (2.13)

Next we discuss the multiplication formula for the multiple generalized 𝑤-Genocchi polynomials as follows: 𝐺^(𝑟)_𝑛,𝑤𝑥;𝑎₁,…,𝑎_𝑟=_ℤ𝑝⋯_ℤ𝑝𝑤^{𝑎1𝑧1+⋯+𝑎𝑟𝑧𝑟}𝑎₁𝑧₁+⋯+𝑎_𝑟𝑧_𝑟+𝑥^𝑛𝑑𝜇₋₁𝑧₁⋯𝑑𝜇₋₁𝑧_𝑟𝑛!=lim_𝑁→∞^{𝑚𝑝𝑁−1}_{𝑧1,…,𝑧𝑟=0}𝑤^{𝑎1𝑧1+⋯+𝑎𝑟𝑧𝑟}𝑎₁𝑧₁+⋯+𝑎_𝑟𝑧_𝑟+𝑥^𝑛(−1)^{𝑧1+⋯+𝑧𝑟}=𝑚^𝑛𝑚−1_{𝑡1,…,𝑡𝑟=0}𝑤^{𝑎1𝑡1+⋯+𝑎𝑟𝑡𝑟}(−1)^{𝑡1+⋯+𝑡𝑟}lim^𝑝_𝑁→∞^𝑁−1_{𝑦1,…,𝑦𝑟=0}(−1)^{𝑚(𝑦1+⋯+𝑦𝑟)}×𝑤^{𝑚𝑎1𝑦1+⋯+𝑎𝑟𝑦𝑟}𝑥+𝑎₁𝑡₁+⋯+𝑎_𝑟𝑡_𝑟𝑚+𝑎₁𝑦₁+⋯+𝑎_𝑟𝑦_𝑟^𝑛𝑛!=𝑚^𝑛𝑚−1_{𝑡1,…,𝑡𝑟=0}𝑤^{𝑎1𝑡1+⋯+𝑎𝑟𝑡𝑟}(−1)^{𝑡1+⋯+𝑡𝑟}𝑛!_ℤ𝑝⋯_ℤ𝑝𝑤^{𝑚𝑎1𝑦1+⋯+𝑎𝑟𝑦𝑟}×𝑥+𝑎₁𝑡₁+⋯+𝑎_𝑟𝑡_𝑟𝑚+𝑎₁𝑦₁+⋯+𝑎_𝑟𝑦_𝑟^𝑛𝑑𝜇₋₁𝑦₁⋯𝑑𝜇₋₁𝑦_𝑟=𝑚^𝑛𝑚−1_{𝑡1,…,𝑡𝑟=0}𝑤^{𝑎1𝑡1+⋯+𝑎𝑟𝑡𝑟}(−1)^{𝑡1+⋯+𝑡𝑟}×𝐺^(𝑟)_{𝑛,𝑤𝑛}𝑥+𝑎₁𝑡₁+⋯+𝑎_𝑟𝑡_𝑟𝑚;𝑎₁,…,𝑎_𝑟. (2.14) Thus, we obtain the following multiplication formula for the multiple generalized 𝑤-Genocchi polynomials.

Theorem 2.4 (multiplication formula). Let 𝑟∈ℕ and 𝑎₁,…,𝑎_𝑟 be strictly positive real numbers. For any 𝑚∈ℕ, one has 𝐺^(𝑟)_𝑛,𝑤𝑚𝑥;𝑎₁,…,𝑎_𝑟=𝑚^𝑛𝑚−1_{𝑡1,…,𝑡𝑟=0}𝑤^{𝑎1𝑡1+⋯+𝑎𝑟𝑡𝑟}(−1)^{𝑡1+⋯+𝑡𝑟}×𝐺^(𝑟)_{𝑛,𝑤𝑛}𝑥+𝑎₁𝑡₁+⋯+𝑎_𝑟𝑡_𝑟𝑚;𝑎₁,…,𝑎_𝑟. (2.15)

Corollary 2.5. (1) If one sets 𝑤=𝑎₁=⋯=𝑎_𝑟=1 and 𝑟,𝑛∈ℕ, then one obtains Raabe type formula for multiple Genocchi polynomials 𝐺_𝑛^(𝑟)(𝑥) as follows: 𝐺_𝑛^(𝑟)(𝑚𝑥)=𝑚^𝑛𝑚−1_{𝑡1,…,𝑡𝑟=0}𝐺_𝑛^(𝑟)𝑥+^𝑛_𝑖=1𝑡_𝑖𝑚, (2.16) where (2𝑡/(𝑒^𝑡+1))^𝑟𝑒^𝑥𝑡=∑^∞_𝑛=0𝐺_𝑛^(𝑟)(𝑥)(𝑡^𝑛/𝑛!).
(2) If one sets 𝑤=1 and 𝑟,𝑛∈ℕ, then one obtains Carlitz's multiplication formula for the multiple generalized Genocchi polynomials 𝐺_𝑛^(𝑟)(𝑥;𝑎₁,…,𝑎_𝑟) as follows: 𝐺_𝑛^(𝑟)𝑚𝑥;𝑎₁,…,𝑎_𝑟=𝑚^𝑛𝑚−1_{𝑡1,…,𝑡𝑟=0}𝐺_𝑛^(𝑟)𝑥+^𝑛_𝑖=1𝑎_𝑖𝑡_𝑖𝑚;𝑎₁,…,𝑎_𝑟, (2.17) where ((2𝑡)^𝑟/(∏^𝑟_𝑗=1(𝑒^𝑎𝑗𝑡+1)))𝑒^𝑥𝑡=∑^∞_𝑛=0𝐺_𝑛^(𝑟)(𝑚𝑥;𝑎₁,…,𝑎_𝑟)(𝑡^𝑛/𝑛!).

Finally, we discuss the recurrence formula for the multiple generalized 𝑤-Genocchi polynomials as follows. Let 𝑟∈ℕ and 𝑎₁,…,𝑎_𝑟 be strictly positive real numbers. For any 𝑘=1,…,𝑟, we can directly derive the following equation: ^∞_𝑛=0^𝑛_𝑗=0𝑛𝑗𝐺^(𝑘)_𝑗,𝑤𝑥∣𝑎₁,…,𝑎_𝑘𝐺^{(𝑟−𝑘)}_{𝑛−𝑗,𝑤}𝑎_𝑘+1,…,𝑎_𝑟𝑡^𝑛=𝑛!_{∞𝑛=0𝑛𝑗=0}𝐺^(𝑘)_𝑗,𝑤𝑥∣𝑎₁,…,𝑎_𝑘𝑡^𝑗𝐺𝑗!^{(𝑟−𝑘)𝑛−𝑗,𝑤}𝑎_𝑘+1,…,𝑎_𝑟𝑡^𝑛−𝑗!=𝑛−𝑗_{∞𝑛=0𝑚+𝑙=𝑛,𝑚,𝑙≥0}𝐺^(𝑘)_𝑚,𝑤𝑥∣𝑎₁,…,𝑎_𝑘𝑡^𝑚𝐺𝑚!^{(𝑟−𝑘)𝑙,𝑤}𝑎_𝑘+1,…,𝑎_𝑟𝑡^𝑙=𝑙!_∞𝑚=0𝐺^(𝑘)_𝑚,𝑤𝑥∣𝑎₁,…,𝑎_𝑘𝑡^𝑚𝑚!^∞_𝑙=0𝐺^{(𝑟−𝑘)}_𝑙,𝑤𝑎_𝑘+1,…,𝑎_𝑟𝑡^𝑙=𝑙!_𝑘𝑗=1(2𝑡)^𝑘(𝑤𝑒^𝑡)^𝑎_𝑗𝑒+1^𝑥𝑡𝑟_𝑗=𝑘+1(2𝑡)^𝑟−𝑘(𝑤𝑒^𝑡)^𝑎_𝑗=+1^𝑟_𝑗=1(2𝑡)^𝑟(𝑤𝑒^𝑡)^𝑎_𝑗𝑒+1^𝑥𝑡=^∞_𝑛=0𝐺_𝑛^(𝑟)𝑚𝑥∣𝑎₁,…,𝑎_𝑟𝑡^𝑛.𝑛! (2.18) By comparing the coefficients on both sides in (2.18), we obtain the recurrence formula for the multiple generalized 𝑤-Genocchi polynomials.

Theorem 2.6 (recurrence formula). Let 𝑟∈ℕ and 𝑎₁,…,𝑎_𝑟 be strictly positive real numbers. For any 𝑘=1,…,𝑟, one has 𝐺_𝑛^(𝑟)𝑚𝑥∣𝑎₁,…,𝑎_𝑟=^𝑛_𝑗=0𝑛𝑗𝐺^(𝑘)_𝑗,𝑤𝑥∣𝑎₁,…,𝑎_𝑘𝐺^{(𝑟−𝑘)}_{𝑛−𝑗,𝑤}𝑎_𝑘+1,…,𝑎_𝑟. (2.19)

Acknowledgment

This paper was supported by Konkuk University in 2011.