Some Identities on the 𝑞-Genocchi Polynomials of Higher-Order and 𝑞-Stirling Numbers by the Fermionic 𝑝-Adic Integral on ℤ𝑝

Abstract

A systemic study of some families of 𝑞-Genocchi numbers and families of polynomials of Nörlund type is presented by using the multivariate fermionic 𝑝-adic integral on ℤ𝑝. The study of these higher-order 𝑞-Genocchi numbers and polynomials yields an interesting 𝑞-analog of identities for Stirling numbers.

1. Introduction

Let 𝑝 be a fixed odd prime number. Throughout this paper, ℤ_𝑝,ℚ_𝑝,ℂ, and ℂ_𝑝 denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, the complex number field, and the completion of the algebraic closure of ℚ_𝑝, respectively. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}. Let 𝑣_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝑣𝑝(𝑝)}=1/𝑝.

When one talks of 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. If 𝑞∈ℂ, then one normally assumes |𝑞|<1. If 𝑞∈ℂ_𝑝, then we assume |𝑞−1|_𝑝<1. In this paper, we use the following notation: [𝑥]_𝑞=1−𝑞^𝑥,[𝑥]1−𝑞^−𝑞=1−−𝑞^𝑥,1+𝑞 (1.1) see [1–10]. Hence lim_𝑞→1[𝑥]_𝑞=𝑥 for all 𝑥∈ℤ_𝑝.

The 𝑞-factorial is defined as [𝑛]_𝑞!=[𝑛]_𝑞[𝑛−1]_𝑞⋯[2]_𝑞[1]_𝑞, and the Gaussian binomial coefficient is defined by the standard rule 𝑛𝑘_𝑞=[𝑛]_𝑞![𝑛−𝑘]_𝑞![𝑘]_𝑞!=[𝑛]_𝑞[𝑛−1]_𝑞⋯[𝑛−𝑘+1]_𝑞![𝑘]_𝑞!, (1.2) (see [7, 9]). Note that lim_𝑞→1(^𝑛_𝑘)_𝑞=(^𝑛_𝑘)=𝑛!/(𝑛−𝑘)!𝑘!=𝑛(𝑛−1)⋯(𝑛−𝑘+1)/𝑘!. It readily follows from (1.2) that 𝑘𝑛+1_𝑞=𝑛𝑘−1_𝑞+𝑞^𝑘𝑛𝑘_𝑞=𝑞^{𝑛−𝑘+1}𝑛𝑘−1_𝑞+𝑛𝑘_𝑞, (1.3) (see [4, 7]).

The 𝑞-binomial formulas are known, 𝑏;𝑞_𝑛=(1−𝑏)1−𝑏𝑞⋯1−𝑏𝑞^𝑛−1=^𝑛_𝑖=0𝑛𝑖_𝑞𝑞^(𝑖2)(−1)^𝑖𝑏^𝑖,1𝑏;𝑞_𝑛=1(1−𝑏)1−𝑏𝑞⋯1−𝑏𝑞^𝑛−1=^∞_𝑖=0𝑖𝑛+𝑖−1_𝑞𝑏^𝑖. (1.4)

We say that 𝑓∶ℤ_𝑝→ℂ_𝑝 is uniformly differentiable function at a point 𝑎∈ℤ_𝑝, and we write 𝑓∈𝑈𝐷(ℤ_𝑝), if the difference quotients Φ_𝑓∶ℤ_𝑝×ℤ_𝑝→ℂ_𝑝 such that Φ_𝑓(𝑥,𝑦)=(𝑓(𝑥)−𝑓(𝑦))/(𝑥−𝑦) have a limit 𝑓^′(𝑎) as (𝑥,𝑦)→(𝑎,𝑎). For 𝑓∈𝑈𝐷(ℤ_𝑝), the 𝑞-deformed fermionic 𝑝-adic integral is defined as 𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)−𝑞^𝑥, (1.5) (see [7, 9]). Note that 𝐼₋₁𝑓=lim_𝑞→1𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥). (1.6) For 𝑛∈ℕ, write 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛). Then, we have 𝐼₋₁𝑓_𝑛=(−1)^𝑛𝐼₋₁𝑓+2^𝑛−1_𝑙=0(−1)^{𝑛−1−𝑙}𝑓(𝑙). (1.7) Using (1.7), we can readily derive the Genocchi polynomials, 𝐺_𝑛(𝑥), namely, 𝑡_ℤ𝑝𝑒^{(𝑥+𝑦)𝑡}𝑑𝜇₋₁𝑦=2𝑡𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐺_𝑛(𝑥)𝑡^𝑛,𝑛! (1.8) (see [1–27]). Note that 𝐺_𝑛(0)=𝐺_𝑛 are referred to as the 𝑛th Genocchi numbers. Let us now introduce the Genocchi polynomials of Nörlund type as follows: 𝑡^𝑟_ℤ𝑝⋯_{ℤ𝑝𝑟times}𝑒^{(𝑥+𝑥1+⋯+𝑥𝑟)𝑡}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟=2𝑡𝑒^𝑡+1^𝑟𝑒^𝑥𝑡=^∞_𝑛=0𝐺_𝑛^(𝑟)(𝑥)𝑡^𝑛,𝑛!𝑒^𝑡+12𝑡^𝑟𝑒^𝑥𝑡=^∞_𝑛=0𝐺_𝑛^(−𝑟)(𝑥)𝑡^𝑛,𝑛! (1.9)(1.10) (see [7, 9]). In the special case 𝑥=0,𝐺_𝑛^(−𝑟)(0)=𝐺_𝑛^(−𝑟), and 𝐺_𝑛^(𝑟)(0)=𝐺_𝑛^(𝑟) are referred to as the Genocchi numbers of Nörlund type. Let (𝐸ℎ)(𝑥)=ℎ(𝑥+1) be the shift operator. Then, the 𝑞-difference operator Δ_𝑞 is defined as Δ^𝑛_𝑞=^𝑛_𝑖=1𝐸−𝑞^𝑖−1𝐼,where(𝐼ℎ)(𝑥)=ℎ(𝑥), (1.11) (see [4, 7, 9]). It follows from (1.11) that 𝑓(𝑥)=_𝑛≥0𝑥𝑛_𝑞Δ^𝑛_𝑞𝑓(0), (1.12) where Δ^𝑛_𝑞𝑓(0)=∑^𝑛_𝑘=0(^𝑛_𝑘)_𝑞𝑞^(𝑘2)𝑓(𝑛−𝑘) (see [5, 6, 10]). The 𝑞-Stirling number of the second kind (as defined by Carlitz) is given by 𝑆₂=𝑞𝑛,𝑘;𝑞^𝑘2)−([𝑘]_𝑞!^𝑘_𝑗=0(−1)^𝑗𝑞^(𝑗2)𝑘𝑗_𝑞𝑘−𝑗^𝑛_𝑞, (1.13) (see [7, 10]). By (1.12) and (1.13), we see that 𝑆₂=𝑞𝑛,𝑘;𝑞^𝑘2)−([𝑘]_𝑞!Δ^𝑘_𝑞0^𝑛, (1.14) (see [6, 10]).

In this paper, the 𝑞-extensions of (1.9) are considered in several ways. Using these 𝑞-extensions, we derive some interesting identities and relations for Genocchi polynomials and numbers of Nörlund type. The purpose of this paper is to present a systemic study of some families 𝑞-Genocchi numbers and polynomials of Nörlund type by using the multivariate fermionic 𝑝-adic integral on ℤ_𝑝.

2.  𝑞-Extensions of Genocchi Numbers and Polynomials of Nörlund Type

In this section, we assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1. We first consider the 𝑞-extensions of (1.8) given by the rule ^∞_𝑛=0𝐺_𝑛,𝑞(𝑥)𝑡^𝑛𝑛!=𝑡_ℤ𝑝𝑒^{[𝑥+𝑦]𝑞𝑡}𝑑𝜇₋₁𝑦=^∞_𝑛=02𝑡1−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥1+𝑞^𝑙𝑡^𝑛𝑛!=2𝑡^∞_𝑚=0(−1)^𝑚𝑒^{[𝑚+𝑥]𝑞𝑡}. (2.1)

Thus, we obtain the following lemma.

Lemma 2.1. If 𝑛≥0, then 𝐺_𝑛+1,𝑞(𝑥)𝑛+1=2^∞_𝑚=0(−1)^𝑚[𝑚+𝑥]^𝑛_𝑞=21−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥1+𝑞^𝑙. (2.2)

By (1.14), [𝑥]^𝑛_𝑞=^𝑛_𝑘=0𝑥𝑘_𝑞[𝑘]_𝑞!𝑆₂𝑘,𝑛−𝑘;𝑞𝑞^(𝑘2)=^𝑛_𝑘=0[𝑥]_𝑞[𝑥−1]_𝑞⋯[𝑥−𝑘+1]_𝑞𝑞^{(𝑘22))−(𝑛−𝑘}[𝑛−𝑘]_𝑞!Δ_𝑞^𝑛−𝑘0^𝑘=^𝑛_𝑘=0𝑞^{(𝑘22))−(𝑛−𝑘}[𝑛−𝑘]_𝑞!Δ_𝑞^𝑛−𝑘0^𝑘11−𝑞^𝑘𝑘_𝑙=0𝑘𝑙_𝑞𝑞^(𝑙2)(−1)^𝑙𝑞^{𝑙(𝑥−𝑘+1)}. (2.3) Thus, we have 𝐺_𝑛+1,𝑞=𝑛+1^𝑛_𝑘=0𝑞^(𝑘2)𝑆₂𝑘,𝑛−𝑘;𝑞1−𝑞^𝑘𝑘_𝑙=0𝑘𝑙_𝑞𝑞^(𝑙2)(−1)^𝑙𝑙_𝑚=0𝑙𝑚𝑞−1^𝑚𝐺_𝑚+1,𝑞(1−𝑘),𝑚+1 (2.4) and we obtain the following theorem.

Theorem 2.2. If 𝑛≥0, then 𝐺_𝑛+1,𝑞=𝑛+1^𝑛_𝑘=0𝑞^(𝑘2)𝑆₂𝑘,𝑛−𝑘;𝑞1−𝑞^𝑘𝑘_𝑙=0𝑘𝑙_𝑞𝑞^(𝑙2)(−1)^𝑙𝑙_𝑚=0𝑙𝑚𝑞−1^𝑚𝐺_𝑚+1,𝑞(1−𝑘),𝑚+1 (2.5) where 𝐺_𝑛,𝑞=𝐺_𝑛,𝑞(0) stand for the nth Genocchi numbers.

Consider a 𝑞-extensoin of (1.9) such that 𝐺^(𝑟)_0,𝑞(𝑥)=𝐺^(𝑟)_1,𝑞(𝑥)=⋯=𝐺^(𝑟)_{𝑟−1,𝑞}(𝑥)=0 and 𝐺^(𝑟)_{𝑛+𝑟,𝑞}(𝑥)𝑟!(_𝑟^𝑛+𝑟)=_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟=2^𝑟1−𝑞^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥11+𝑞^𝑙𝑟=2^𝑟∞_𝑚=0𝑚()𝑚+𝑟−1−1^𝑚[𝑚+𝑥]^𝑛_𝑞. (2.6) Let 𝐹_𝑞^(𝑟)(𝑡,𝑥)=∑^∞_𝑛=0𝐺^(𝑟)_𝑛,𝑞(𝑥)(𝑡^𝑛/𝑛!). Then, 𝐹_𝑞^(𝑟)(𝑡,𝑥)=2^𝑟𝑡^𝑟∞_𝑚=0𝑚()𝑚+𝑟−1−1^𝑚𝑒^{[𝑚+𝑥]𝑞𝑡}. (2.7) In the special case 𝑥=0, the numbers 𝐺^(𝑟)_𝑛,𝑞(0)=𝐺^(𝑟)_𝑛,𝑞 are referred to as 𝑞-extension of the Genocchi numbers of order 𝑟. In the sense of the 𝑞-extension in (1.10), consider the 𝑞-extension of Genocchi polynomials of Nörlund type given by 𝐺_𝑞^(𝑟)(𝑡,𝑥)=𝐹_𝑞^(−𝑟)()=1𝑡,𝑥2^𝑟𝑡^𝑟𝑟_𝑚=0𝑟𝑚𝑒^{[𝑚+𝑥]𝑞𝑡}=^∞_𝑛=0𝐺^(−𝑟)_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (2.8) By (2.8), 𝐺^(−𝑟)_0,𝑞(𝑥)=𝐺^(−𝑟)_1,𝑞(𝑥)=⋯=𝐺^(−𝑟)_{𝑟−1,𝑞}(𝑥)=0 and 𝑟!(^𝑛_𝑟)𝐺^(𝑟)_{𝑛−𝑟,𝑞}(𝑥)=(1/2^𝑟)∑^𝑟_𝑚=0(^𝑟_𝑚)[𝑚+𝑥]^𝑛_𝑞. Therefore, we obtain the following theorem.

Theorem 2.3. For 𝑟∈ℕ, and, 𝑛≥0, write 2^𝑟𝑡^𝑟∞_𝑚=0𝑚()𝑚+𝑟−1−1^𝑚𝑒^{[𝑚+𝑥]𝑞𝑡}=^∞_𝑛=0𝐺^(𝑟)_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (2.9) Then, 𝐺^(𝑟)_𝑛+r,𝑞(𝑥)𝑟!(_𝑟^𝑛+𝑟)=2^𝑟1−𝑞^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥11+𝑞^𝑙𝑟=2^𝑟∞_𝑚=0𝑚()𝑚+𝑟−1−1^𝑚[𝑚+𝑥]^𝑛_𝑞,𝑛𝑟𝑟!𝐺^(−𝑟)_{𝑛−𝑟,𝑞}(𝑥)=12^𝑟1−𝑞^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥1+𝑞^𝑙𝑟=12^𝑟𝑟_𝑚=0𝑟𝑚[𝑚+𝑥]^𝑛_𝑞. (2.10)

The numbers 𝐺^(−𝑟)_𝑛,𝑞(0)=𝐺^(−𝑟)_𝑛,𝑞 are referred to as the 𝑞-extension of Genocchi numbers of Nörlund type. For ℎ∈ℤ and 𝑟∈ℕ, introduce the extended higher-order 𝑞-Genocchi polynomials as follows: 𝐺^(ℎ,𝑟)_{𝑛+𝑟,𝑞}(𝑥)𝑟!(_𝑟^𝑛+𝑟)=_ℤ𝑝⋯_ℤ𝑝𝑞^{∑𝑟𝑗=1(ℎ−𝑗)𝑥𝑗}𝑥+𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟. (2.11) Then, 𝐺^(ℎ,𝑟)_{𝑛+𝑟,𝑞}(𝑥)𝑟!(_𝑟^𝑛+𝑟)=2^𝑟1−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥−𝑞^ℎ−1+𝑙;𝑞⁻¹_𝑟=2^𝑟1−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥−𝑞^{ℎ−𝑟+𝑙};𝑞_𝑟=2^𝑟∞_𝑚=0𝑚𝑚+𝑟−1_𝑞(−1)^𝑚𝑞^{(ℎ−𝑟)𝑚}[𝑥+𝑚]^𝑛_𝑞. (2.12) Let 𝐹_𝑞^(ℎ,𝑟)(𝑡,𝑥)=∑^∞_𝑛=0𝐺^(ℎ,𝑟)_𝑛,𝑞(𝑥)(𝑡^𝑛/𝑛!). Then, we can readily see that 𝐹_𝑞^(ℎ,𝑟)(𝑡,𝑥)=2^𝑟𝑡^𝑟∞_𝑚=0𝑚𝑚+𝑟−1_𝑞(−1)^𝑚𝑞^{(ℎ−𝑟)𝑚}𝑒^{[𝑥+𝑚]𝑞𝑡}. (2.13) Therefore, we obtain the following theorem.

Theorem 2.4. For ℎ∈ℤ and 𝑛≥0, let 2^𝑟𝑡^𝑟∞_𝑚=0𝑚𝑚+𝑟−1_𝑞(−1)^𝑚𝑞^{(ℎ−𝑟)𝑚}𝑒^{[𝑥+𝑚]𝑞𝑡}=^∞_𝑛=0𝐺^(ℎ,𝑟)_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (2.14) Then, 𝐺^(ℎ,𝑟)_{𝑛+𝑟,𝑞}(𝑥)𝑟!(_𝑟^𝑛+𝑟)=2^𝑟1−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥−𝑞^{ℎ−𝑟+𝑙};𝑞_𝑟=2^𝑟∞_𝑚=0𝑚𝑚+𝑟−1_𝑞(−1)^𝑚𝑞^{(ℎ−𝑟)𝑚}[𝑥+𝑚]^𝑛_𝑞. (2.15)

Let us now define the extended higher-order Nörlund type 𝑞-Genocchi polynomials as follows: 𝑛𝑟𝑟!𝐺^{(ℎ,−𝑟)}_{𝑛−𝑟,𝑞}(𝑥)=11−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥∫_ℤ𝑝⋯∫_ℤ𝑝𝑞^{𝑙(𝑥1+⋯+𝑥𝑟)}𝑞^{∑𝑟𝑗=1(ℎ−𝑗)𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟. (2.16) By (2.16), 𝑛𝑟𝑟!𝐺^{(ℎ,−𝑟)}_{𝑛−𝑟,𝑞}(𝑥)=12^𝑟1−𝑞^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥−𝑞^{ℎ−𝑟+𝑙};𝑞_𝑟=12^𝑟𝑟_𝑚=0𝑟𝑚_𝑞𝑞^(𝑚2)𝑞^{(ℎ−𝑟)𝑚}[𝑚+𝑥]^𝑛_𝑞. (2.17) Let 𝐹_𝑞^{(ℎ,−𝑟)}(𝑡,𝑥)=∑^∞_𝑛=0𝐺^{(ℎ,−𝑟)}_𝑛,𝑞(𝑥)(𝑡^𝑛/𝑛!). Then, we have 𝐹_𝑞^{(ℎ,−𝑟)}()=1𝑡,𝑥2^𝑟𝑡^𝑟𝑟_𝑚=0𝑟𝑚_𝑞𝑞^(𝑚2)𝑞^{(ℎ−𝑟)𝑚}𝑒^{[𝑚+𝑥]𝑞𝑡}, (2.18) where, 𝐺^{(ℎ,−𝑟)}_0,𝑞(𝑥)=𝐺^{(ℎ,−𝑟)}_1,𝑞(𝑥)=⋯=𝐺^{(ℎ,−𝑟)}_{𝑟−1,𝑞}(𝑥)=0. Therefore, we obtain the following theorem.

Theorem 2.5. For ℎ∈ℤ, 𝑛≥0, and 𝑟∈ℕ, write 12^𝑟𝑡^𝑟𝑟_𝑚=0𝑟𝑚_𝑞𝑞^(𝑚2)𝑞^{(ℎ−𝑟)𝑚}𝑒^{[𝑚+𝑥]𝑞𝑡}=^∞_𝑛=0𝐺^{(ℎ,−𝑟)}_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (2.19) Then, 𝑛𝑟𝑟!𝐺^{(ℎ,−𝑟)}_{𝑛−𝑟,𝑞}(𝑥)=12^𝑟1−𝑞^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥−𝑞^{ℎ−𝑟+𝑙};𝑞_𝑟=12^𝑟𝑟_𝑚=0𝑟𝑚_𝑞𝑞^(𝑚2)𝑞^{(ℎ−𝑟)𝑚}[𝑚+𝑥]^𝑛_𝑞, (2.20) where, 𝐺^{(ℎ,−𝑟)}_0,𝑞(𝑥)=𝐺^{(ℎ,−𝑟)}_1,𝑞(𝑥)=⋯=𝐺^{(ℎ,−𝑟)}_{𝑟−1,𝑞}(𝑥)=0.

For ℎ=𝑟, 𝐺^(𝑟,𝑟)_{𝑛+𝑟,𝑞}(𝑥)𝑟!(_𝑟^𝑛+𝑟)=2^𝑟1−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥−𝑞^𝑙;𝑞_𝑟=2^𝑟∞_𝑚=0𝑚𝑚+𝑟−1_𝑞(−1)^𝑚[𝑥+𝑚]^𝑛_𝑞,𝑛𝑟𝑟!𝐺^{(𝑟,−𝑟)}_{𝑛−𝑟,𝑞}(𝑥)=12^𝑟1−𝑞^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥−𝑞^𝑙;𝑞_𝑟=12^𝑟𝑟_𝑚=0𝑟𝑚_𝑞𝑞^(𝑚2)[𝑚+𝑥]^𝑛_𝑞. (2.21)(2.22) It can readily be seen that 𝑞^𝑚𝑥2^𝑟−𝑞^𝑚−𝑟;𝑞_𝑟=_ℤ𝑝⋯_ℤ𝑝𝑞^{∑𝑟𝑗=1(𝑚−𝑗)𝑥𝑗+𝑚𝑥}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟=_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑟𝑞𝑞−1+1^𝑚𝑞^{−∑𝑟𝑗=1𝑗𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟=^𝑚_𝑙=0𝑚𝑙𝑞−1^𝑙_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑟^𝑙_𝑞𝑞^{−∑𝑟𝑗=1𝑗𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟=^𝑚_𝑙=0𝑚𝑙𝑞−1^𝑙𝐺^(0,𝑟)_{𝑙+𝑟,𝑞}(𝑥)𝑟!_𝑟^𝑙+𝑟. (2.23) By (2.23), 𝑞^𝑚𝑥2^𝑟/(−𝑞^𝑚−𝑟;𝑞)_𝑟=∑^𝑚_𝑙=0(^𝑚_𝑙)(𝑞−1)^𝑙(𝐺^(0,𝑟)_{𝑙+𝑟,𝑞}(𝑥)/𝑟!(_𝑟^𝑙+𝑟)). As is known, 𝐼₋₁𝑓₁+𝐼₋₁𝑓=2𝑓(0),where𝑓₁(𝑥)=𝑓(𝑥+1). (2.24) It follows from (2.24) that 𝑞^ℎ−1_ℤ𝑝⋯_ℤ𝑝𝑥+1+𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝑞^{∑𝑟𝑗=1(ℎ−𝑗)𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟=−_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝑞^{∑𝑟𝑗=1(ℎ−𝑗)𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟+2_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₂+⋯+𝑥_𝑟^𝑛_𝑞𝑞^{∑𝑟−1𝑗=1(ℎ−1−𝑗)𝑥𝑗+1}𝑑𝜇₋₁𝑥₂⋯𝑑𝜇₋₁𝑥_𝑟. (2.25) By (2.25), 𝑞^ℎ−1𝐺^(ℎ,𝑟)_{𝑛+𝑟,𝑞}(𝑥+1)+𝐺𝑛+𝑟^{(ℎ,𝑟)𝑛+𝑟,𝑞}(𝑥)𝑛+𝑟=2𝐺^{(ℎ−1,𝑟−1)}_{𝑛+𝑟−1,𝑞}(𝑥). (2.26) A simple manipulation shows that 𝑞^𝑥_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝑞^{∑𝑟𝑗=1(ℎ−𝑗+1)𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟=𝑞−1_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑟𝑞^𝑛+1𝑞^{∑𝑟𝑗=1(ℎ−𝑗)𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟+_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝑞^{∑𝑟𝑗=1(ℎ−𝑗)𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟. (2.27) By (2.27), 𝑞^𝑥(𝐺^(ℎ+1,𝑟)_{𝑛+𝑟,𝑞}(𝑥)/(𝑛+1))=(𝑞−1)(𝐺^(ℎ,𝑟)_{𝑛+𝑟+1,𝑞}(𝑥)/(𝑛+𝑟+1))+(𝐺^(ℎ,𝑟)_{𝑛+𝑟,𝑞}(𝑥)/(𝑛+1)).

Therefore, we obtain the following proposition.

Proposition 2.6. For ℎ∈ℤ, 𝑟∈ℕ and 𝑛≥0, the following equations 𝑞^ℎ−1𝐺^(ℎ,𝑟)_n+𝑟,𝑞(𝑥+1)+𝐺𝑛+𝑟^{(ℎ,𝑟)𝑛+𝑟,𝑞}(𝑥)𝑛+𝑟=2𝐺^{(ℎ−1,𝑟−1)}_{𝑛+𝑟−1,𝑞}(𝑥),𝑞^𝑥𝐺^(ℎ+1,𝑟)_{𝑛+𝑟,𝑞}(𝑥)=𝐺𝑛+1𝑞−1^{(ℎ,𝑟)𝑛+𝑟+1,𝑞}(𝑥)+𝐺𝑛+𝑟+1^{(ℎ,𝑟)𝑛+𝑟,𝑞}(𝑥)𝑛+1 (2.28) hold. Moreover, (𝑞^𝑚𝑥2^𝑟)/((−𝑞^𝑚−𝑟;𝑞)_𝑟)=∑^𝑚_𝑙=0(^𝑚_𝑙)(𝑞−1)^𝑙(𝐺^(0,𝑟)_l+𝑟,𝑞(𝑥)/𝑟!(_𝑟^𝑙+𝑟)).

By (2.21), 𝐺^(𝑟,𝑟)_{𝑛+𝑟,𝑞−1}(𝑟−𝑥)𝑟!(_𝑟^𝑛+𝑟)=2^𝑟1−𝑞^−1𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^{−𝑙(𝑟−𝑥)}−𝑞^−𝑙;𝑞⁻¹_𝑟=(−1)^𝑛𝑞^𝑟2)𝑛+(2^𝑟1−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑥−𝑞^𝑙;𝑞_𝑟=(−1)^𝑛𝑞^𝑟2)𝑛+(𝐺^(𝑟,𝑟)_{𝑛+𝑟,𝑞}(𝑥)𝑟!(_𝑟^𝑛+𝑟). (2.29) Hence, _ℤ𝑝⋯_ℤ𝑝𝑟−𝑥+𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞−1𝑞^{−∑𝑟𝑗=1(𝑟−𝑗)𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟=(−1)^𝑛𝑞^𝑟2)𝑛+(_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝑞^{∑𝑟𝑗=1(𝑟−𝑗)𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟. (2.30) For ℎ=𝑟, 𝐺^(𝑟,𝑟)_{𝑛+𝑟,𝑞−1}(0)=(−1)^𝑛𝑞^𝑟2)𝑛+(𝐺^(𝑟,𝑟)_{𝑛+𝑟,𝑞}(𝑟). It also follows from (2.26) that 𝑞^𝑟−1𝐺^(𝑟,𝑟)_{𝑛+𝑟,𝑞}(𝑥+1)+𝐺𝑛+𝑟^{(𝑟,𝑟)𝑛+𝑟,𝑞}(𝑥)𝑛+𝑟=2𝐺^{(𝑟−1,𝑟−1)}_{𝑛+𝑟−1,𝑞}(𝑥). (2.31)

The Stirling numbers of the first kind are defined as 1+[^𝑛_𝑘=1𝑘]_𝑞𝑧=^𝑛_𝑘=0𝑆₁𝑛,𝑘;𝑞𝑧^𝑘, (2.32) (see[6, 9]), 𝑞^(𝑚2)𝑟𝑚_𝑞=𝑞^(𝑚2)[𝑟]_𝑞⋯[𝑟−𝑚+1]_𝑞[𝑚]_𝑞!=1[𝑚]_𝑞!^𝑚−1_𝑘=0[𝑟]_𝑞−[𝑘]_𝑞. (2.33) It can readily be seen that 𝑧−[^𝑛−1_𝑘=0𝑘]_𝑞=𝑧^𝑛𝑛−1_𝑘=0[𝑘]1−^𝑞𝑧=^𝑛_𝑘=0𝑆₁𝑛−1,𝑘;𝑞(−1)^𝑘𝑧^𝑛−𝑘. (2.34) By (2.33) and (2.34), ^𝑚−1_𝑘=0[𝑟]_𝑞−[𝑘]_𝑞=^𝑚_𝑘=0𝑆₁𝑚−1,𝑘;𝑞(−1)^𝑘[𝑟]_𝑞^𝑚−𝑘. (2.35) Formulas (2.22) and (2.35) imply the following assertion.

Proposition 2.7. For 𝑟∈ℕ and 𝑛∈ℤ₊, 𝑛𝑟𝑟!𝐺^{(𝑟,−𝑟)}_{𝑛−𝑟,𝑞}(𝑥)=12^𝑟[𝑚]_𝑞!^𝑟𝑚_𝑚=0𝑘=0𝑆₁𝑚−1,𝑘;𝑞(−1)^𝑘[𝑟]_𝑞^𝑚−𝑘[𝑚+𝑥]^𝑛_𝑞. (2.36)

The generalized Genocchi numbers and polynomials of Nörlund type are defined by 2^𝑟𝑡^𝑟(𝑒^𝑤1𝑡+1)(𝑒^𝑤2𝑡+1)⋯(𝑒^𝑤𝑟𝑡)𝑒+1^𝑥𝑡=^∞_𝑛=0𝐺_𝑛^(𝑟)𝑥∣𝑤₁,…,𝑤_𝑟𝑡^𝑛,𝑛! (2.37) and 𝐺_𝑛^(𝑟)(𝑤₁,…,𝑤_𝑟)=𝐺_𝑛^(𝑟)(0∣𝑤₁,…,𝑤_𝑟). We can now also define a 𝑞-extension of (2.37) as follows. For 𝑤₁,…,𝑤_𝑟∈ℤ_𝑝 and 𝛿₁,…,𝛿_𝑟∈ℤ, write 𝐺^(𝑟)_{𝑛+𝑟,𝑞}𝑥∣𝑤₁,…,𝑤_𝑟;𝛿₁,…,𝛿_𝑟𝑟!(_𝑟^𝑛+𝑟)=_ℤ𝑝⋯_ℤ𝑝𝑥₁𝑤₁+⋯+𝑥_𝑟𝑤_𝑟+𝑥^𝑛_𝑞𝑑𝜇_−𝑞𝛿1𝑥₁⋯𝑑𝜇_{−𝑞𝛿𝑟}𝑥_𝑟, (2.38)

and 𝐺^(𝑟)_{𝑛+𝑟,𝑞}(𝑤₁,…,𝑤_𝑟;𝛿₁,…,𝛿_𝑟)=𝐺^(𝑟)_{𝑛+𝑟,𝑞}(0∣𝑤₁,…,𝑤_𝑟;𝛿₁,…,𝛿_𝑟). Thus, 𝐺^(𝑟)_{𝑛+𝑟,𝑞}𝑥∣𝑤₁,…,𝑤_𝑟;𝛿₁,…,𝛿_𝑟𝑟!(_𝑟^𝑛+𝑟)=[2]_𝑞𝛿1⋯[2]_𝑞𝛿𝑟1−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥1+𝑞^{𝛿1+𝑙𝑤1}⋯1+𝑞^{𝛿𝑟+𝑙𝑤𝑟}. (2.39) Another 𝑞-extension of Nörlund type generalized Genocchi numbers and polynomials is also of interest, namely, 𝐺^∗(𝑟)_{𝑛+𝑟,𝑞}𝑥∣𝑤₁,…,𝑤_𝑟;𝛿₁,…,𝛿_𝑟𝑟!(_𝑟^𝑛+𝑟)=_ℤ𝑝⋯_ℤ𝑝𝑥₁𝑤₁+⋯+𝑥_𝑟𝑤_𝑟+𝑥^𝑛_𝑞𝑞^{𝛿1𝑥1+⋯+𝛿𝑟𝑥𝑟}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟, (2.40) and 𝐺^∗(𝑟)_{𝑛+𝑟,𝑞}(𝑤₁,…,𝑤_𝑟;𝛿₁,…,𝛿_𝑟)=𝐺^∗(𝑟)_{𝑛+𝑟,𝑞}(0∣𝑤₁,…,𝑤_𝑟;𝛿₁,…,𝛿_𝑟). By (2.40), 𝐺^∗(𝑟)_{𝑛+𝑟,𝑞}𝑥∣𝑤₁,…,𝑤_𝑟;𝛿₁,…,𝛿_𝑟𝑟!(_𝑟^𝑛+𝑟)=2^𝑟1−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥1+𝑞^{𝛿1+𝑙𝑤1}⋯1+𝑞^{𝛿𝑟+𝑙𝑤𝑟}. (2.41)

3. Further Remarks

For ℎ=0, consider the following polynomials 𝐺^(0,𝑟)_{𝑛+𝑟,𝑞}(𝑥)/𝑟!(_𝑟^𝑛+𝑟) and 𝑟!(^𝑛_𝑟)𝐺^(0,−𝑟)_{𝑛+𝑟,𝑞}(𝑥): 𝐺^(0,𝑟)_{𝑛+𝑟,𝑞}(𝑥)𝑟!(_𝑟^𝑛+𝑟)=_ℤ𝑝⋯_ℤ𝑝𝑥+𝑥₁+⋯+𝑥_𝑟^𝑛_𝑞𝑞^{−∑𝑟𝑗=1𝑗𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟,𝑛𝑟𝑟!𝐺^(0,−𝑟)_{𝑛−𝑟,𝑞}(𝑥)=11−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥∫_ℤ𝑝⋯∫_ℤ𝑝𝑞^{𝑙(𝑥1+⋯+𝑥𝑟)}𝑞^{−∑𝑟𝑗=1𝑗𝑥𝑗}𝑑𝜇₋₁𝑥₁⋯𝑑𝜇₋₁𝑥_𝑟. (3.1) Then, 𝐺^(0,𝑟)_{𝑛+𝑟,𝑞}(𝑥)𝑟!(_𝑟^𝑛+𝑟)=2^𝑟1−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥−𝑞^𝑙−𝑟;𝑞_𝑟=2^𝑟∞_𝑚=0𝑚𝑚+𝑟−1_𝑞𝑞^−𝑟𝑚(−1)^𝑚[𝑥+𝑚]^𝑛_𝑞𝑛𝑟𝑟!𝐺^(0,−𝑟)_{𝑛−𝑟,𝑞}(𝑥)=12^𝑟1−𝑞^𝑛𝑛_𝑙=0𝑛𝑙(−1)^𝑙𝑞^𝑙𝑥−𝑞^𝑙−𝑟;𝑞_𝑟=12^𝑟𝑟_𝑚=0𝑟𝑚_𝑞𝑞^(𝑚2)𝑞^−𝑟𝑚[𝑚+𝑥]^𝑛_𝑞. (3.2) Let 𝐹_𝑞^(0,𝑟)(𝑡,𝑥)=∑^∞_𝑛=0𝐺^(0,𝑟)_𝑛,𝑞(𝑥)(𝑡^𝑛/𝑛!) and let 𝐹_𝑞^(0,−𝑟)(𝑡,𝑥)=∑^∞_𝑛=0𝐺^(0,−𝑟)_𝑛,𝑞(𝑥)(𝑡^𝑛/𝑛!). Then, 𝐹_𝑞^(0,𝑟)(𝑡,𝑥)=2^𝑟𝑡^𝑟∞_𝑚=0𝑚𝑚+𝑟−1_𝑞𝑞^−𝑟𝑚(−1)^𝑚𝑒^{[𝑥+𝑚]𝑞𝑡},𝐹_𝑞^(0,−𝑟)()=1𝑡,𝑥2^𝑟𝑡^𝑟𝑟_𝑚=0𝑟𝑚_𝑞𝑞^(𝑚2)𝑞^−𝑟𝑚𝑒^{[𝑚+𝑥]𝑞𝑡}. (3.3) Consider the following polynomials: 𝐺^(ℎ,1)_𝑛+1,𝑞(𝑥)=𝑛+1_ℤ𝑝𝑞^{𝑥1(ℎ−1)}𝑥+𝑥₁^𝑛_𝑞𝑑𝜇₋₁𝑥₁=21−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥1+𝑞^𝑙+ℎ−1. (3.4) A simple calculation of the fermionic 𝑝-adic invariant integral on ℤ_𝑝 show that 𝑞^𝑥_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝑞^{𝑥1(ℎ−1)}𝑑𝜇₋₁𝑥₁=𝑞−1_ℤ𝑝𝑥+𝑥_1𝑞^𝑛+1𝑞^{𝑥1(ℎ−2)}𝑑𝜇₋₁𝑥₁+_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝑞^{𝑥1(ℎ−2)}𝑑𝜇₋₁𝑥₁. (3.5) By (3.5), 𝑞^𝑥𝐺^(ℎ,1)_𝑛+1,𝑞(𝑥)=(𝑞−1)(𝐺^(ℎ−1,1)_𝑛+2,𝑞(𝑥)/2(𝑛+2))+𝐺^(ℎ−1,1)_𝑛+1,𝑞(𝑥). It can readily be proved that _ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝑞^{𝑥1(ℎ−1)}𝑑𝜇₋₁𝑥₁=^𝑛_𝑗=0𝑛𝑗[𝑥]_𝑞^𝑛−𝑗𝑞^𝑗𝑥_ℤ𝑝𝑥₁^𝑗_𝑞𝑞^{𝑥1(ℎ−1)}𝑑𝜇₋₁𝑥₁. (3.6) By (3.6), 𝐺^(ℎ,1)_𝑛+1,𝑞(𝑥)/(𝑛+1)=∑^𝑛_𝑗=0(^𝑛_𝑗)[𝑥]_𝑞^𝑛−𝑗𝑞^𝑗𝑥(𝐺^(ℎ,1)_𝑗+1,𝑞/(𝑗+1)). Using (2.24), we can also prove that _ℤ𝑝𝑥+𝑥₁+1^𝑛_𝑞𝑞^{(𝑥1+1)(ℎ−1)}𝑑𝜇₋₁𝑥₁+_ℤ𝑝𝑥+𝑥₁^𝑛_𝑞𝑞^{𝑥1(ℎ−1)}𝑑𝜇₋₁𝑥₁=2[𝑥]^𝑛_𝑞. (3.7) Thus, 𝑞^ℎ−1(𝐺^(ℎ,1)_𝑛+1,𝑞(𝑥)/(𝑛+1))+(𝐺^(ℎ,1)_𝑛+1,𝑞(𝑥)/(𝑛+1))=2[𝑥]^𝑛_𝑞. For 𝑥=0, we have 𝑞^ℎ−1(𝐺^(ℎ,1)_𝑛+1,𝑞(1)/(𝑛+1))+(𝐺^(ℎ,1)_𝑛+1,𝑞/(𝑛+1))=2𝛿_𝑛,0, where 𝛿_𝑛,0 is the Kronecker delta.

It is easy to see that 𝐺^(ℎ,1)_1,𝑞=∫_ℤ𝑝𝑞^{𝑥1(ℎ−1)}𝑑𝜇₋₁(𝑥₁)=2/(1+𝑞^ℎ−1)=2/([2]_𝑞ℎ−1). By (3.4), 𝐺^(ℎ,1)_{𝑛+1,𝑞−1}(1−𝑥)=𝑛+1_ℤ𝑝1−𝑥+𝑥₁^𝑛_𝑞−1𝑞^{−𝑥1(ℎ−1)}𝑑𝜇₋₁𝑥₁=(−1)^𝑛𝑞^𝑛+ℎ−121−𝑞^𝑛𝑛_𝑙=0(^𝑛_𝑙)(−1)^𝑙𝑞^𝑙𝑥1+𝑞^𝑙+ℎ−1=(−1)^𝑛𝑞^𝑛+ℎ−1𝐺^(ℎ,1)_𝑛+1,𝑞(𝑥).𝑛+1 (3.8) In particular, if 𝑥=1, then 𝐺^(ℎ,1)_{𝑛+1,𝑞−1}(0)/(𝑛+1)=(−1)^𝑛𝑞^𝑛+ℎ−1(𝐺^(ℎ,1)_𝑛+1,𝑞(1)/(𝑛+1))=(−1)^𝑛−1𝑞^𝑛(G^(ℎ,1)_𝑛+1,𝑞/(𝑛+1)) for 𝑛≥1.

Recently, Kim has studied 𝑝-adic fermionic integral on ℤ_𝑝 connected with the problems of mathematical physics (see [6, 10, 11]), and our result are closely related to his results. In the future, we will try to study 𝑝-adic stochastic problems associated with our theorems. For example, 𝑝-adic 𝑞-Bernstein polynomials seem to be closely related to our results (see [6, 14, 20]).