Integral Formulae of Bernoulli and Genocchi Polynomials

Abstract

Recently, some interesting and new identities are introduced in the work of Kim et al. (2012). From these identities, we derive some new and interesting integral formulae for Bernoulli and Genocchi polynomials.

1. Introduction

As it is well known, the Bernoulli polynomials are defined by generating functions as follows: 𝑡𝑒^𝑡𝑒−1^𝑥𝑡=𝑒^{𝐵(𝑥)𝑡}=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛𝑛! (1.1) (see [1–5]) with the usual convention about replacing 𝐵^𝑛(𝑥) by 𝐵_𝑛(𝑥). In the special case, 𝑥=0, 𝐵_𝑛(0)=𝐵_𝑛 are called the 𝑛th Bernoulli numbers.

The Genocchi polynomials are also defined by 2𝑡𝑒^𝑡𝑒+1^𝑥𝑡=𝑒^{𝐺(𝑥)𝑡}=^∞_𝑛=0𝐺_𝑛(𝑥)𝑡^𝑛𝑛! (1.2) (see [1, 6–10]) with the usual convention about replacing 𝐺^𝑛(𝑥) by 𝐺_𝑛(𝑥). In the special case, 𝑥=0, 𝐺_𝑛(0)=𝐺_𝑛 are called the 𝑛th Genocchi numbers.

From (1.1), we note that 𝐵_𝑛(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐵_𝑙𝑥^𝑛−𝑙 (1.3) (see [1–5]). Thus, by (1.3), we get 𝑑𝐵𝑑𝑥^𝑛(𝑥)=𝑛^𝑛−1_𝑙=0𝑙𝐵𝑛−1_𝑙𝑥^{𝑛−1−𝑙}=𝑛𝐵_𝑛−1(𝑥) (1.4) (see [2]). From (1.2), we note that 𝐺_𝑛(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐺_𝑙𝑥^𝑛−𝑙. (1.5) From (1.5), we can derive the following equation: 𝑑𝐺𝑑𝑥^𝑛(𝑥)=𝑛^𝑛−1_𝑙=0𝑙𝐺𝑛−1_𝑙𝑥^{𝑛−1−𝑙}=𝑛𝐺_𝑛−1(𝑥). (1.6) By the definition of Bernoulli and Genocchi numbers, we get the following recurrence formulae: 𝐵₀=1,𝐵_𝑛(1)−𝐵_𝑛=𝛿_1,𝑛,𝐺₀=0,𝐺_𝑛(1)+𝐺_𝑛=2𝛿_1,𝑛, (1.7) where 𝛿_𝑛,𝑘 is the Kronecker symbol (see [2]). From (1.4), (1.6), and (1.7), we note that ¹₀𝐵_𝑛(𝑥)𝛿𝑑𝑥=^0,𝑛(),𝑛+1𝑛≥0¹₀𝐺_𝑛(𝑥)𝑑𝑥=−2𝐺_𝑛+1().𝑛+1𝑛≥1 (1.8) From the identities of Bernoulli and Genocchi polynomials, we derive some new and interesting integral formulae of an arithmetical nature on the Bernoulli and Genocchi polynomials.

2. Integral Formula of Bernoulli and Genocchi Polynomials

From (1.1) and (1.2), we note that 𝑡𝑒^𝑡𝑒−1^𝑥𝑡=122𝑡𝑒^𝑥𝑡𝑒^𝑡+1+1𝑡𝑡𝑒^𝑡−12𝑡𝑒^𝑥𝑡𝑒^𝑡=1+12^∞_𝑛=0𝐺_𝑛(𝑥)𝑡^𝑛+1𝑛!𝑡^∞_𝑙=0𝐵_𝑙𝑡^𝑙𝑙!^∞_𝑚=0𝐺_𝑚(𝑥)𝑡^𝑚=1𝑚!2^∞_𝑛=0𝐺_𝑛(𝑥)𝑡^𝑛+1𝑛!𝑡^∞𝑛_𝑛=0𝑙=0𝑛𝑙𝐺_𝑙(𝑥)𝐵_𝑛−𝑙𝑡^𝑛=1𝑛!2^∞_𝑛=0𝐺_𝑛(𝑥)𝑡^𝑛+𝑛!^∞_𝑛=0−12𝐺_𝑛(𝑥)+^𝑛+1_{𝑙=0𝑙≠𝑛}^𝑙𝑛+1𝐺_𝑙(𝑥)𝐵_{𝑛+1−𝑙}𝑡𝑛+1^𝑛=𝑛!_{∞𝑛=0𝑛+1𝑙=0𝑙≠𝑛}𝑙𝑛+1𝐺_𝑙(𝑥)𝐵_{𝑛+1−𝑙}𝑡𝑛+1^𝑛.𝑛! (2.1) By comparing the coefficients on the both sides of (2.1), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊, one has 𝐵_𝑛(𝑥)=^𝑛+1_{𝑙=0𝑙≠𝑛}𝑙𝑛+1𝐺_𝑙(𝑥)𝐵_{𝑛+1−𝑙}𝑛+1. (2.2)

From (1.1) and (1.2), also notes that 2𝑡𝑒^𝑡𝑒+1^𝑥𝑡=1𝑡2𝑡𝑒^𝑡−1𝑒^𝑡+1𝑡𝑒^𝑥𝑡𝑒^𝑡=1−1𝑡2𝑡−22𝑡𝑒^𝑡+1𝑡𝑒^𝑥𝑡𝑒^𝑡=1−1𝑡2𝑡−2^∞_𝑙=0𝐺_𝑙𝑡^𝑙𝑙!^∞_𝑚=0𝐵_𝑚(𝑥)𝑡^𝑚=1𝑚!𝑡−2^∞_𝑙=1𝐺_𝑙+1𝑡𝑙+1𝑙!^𝑙+1∞_𝑚=0𝐵_𝑚(𝑥)𝑡^𝑚=𝑚!_∞𝑛=1−2^𝑛_𝑙=1𝑛𝑙𝐺_𝑙+1𝐵𝑙+1^𝑛−𝑙(𝑥)𝑡^𝑛.𝑛! (2.3) By comparing the coefficients on the both sides of (2.3), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℕ, one has 𝐺_𝑛(𝑥)=−2^𝑛_𝑙=1𝑛𝑙𝐺_𝑙+1𝐵𝑙+1^𝑛−𝑙(𝑥). (2.4)

Let one take the definite integral from 0 to 1 on both sides of Theorem 2.1. For 𝑛≥2, 0=−2^𝑛+1_{𝑙=1𝑙≠𝑛}𝑙𝑛+1𝐺_𝑙+1𝐵𝑙+1𝑛+1^{𝑛+1−𝑙}=−𝐵_𝑛𝐺₂−2^𝑛_{𝑙=1𝑙≠𝑛−1}𝑛𝑙𝐵_𝑛−𝑙𝐺_𝑙+2().𝑙+1)(𝑙+2 (2.5) Therefore, by (2.3), we obtain the following theorem.

Theorem 2.3. For 𝑛∈ℕ with 𝑛≥2, one has 𝐵_𝑛=2^𝑛_{𝑙=1𝑙≠𝑛−1}𝑛𝑙𝐵_𝑛−𝑙𝐺_𝑙+2().𝑙+1)(𝑙+2 (2.6)

3. 𝑝-Adic Integral on ℤ_𝑝 Associated with Bernoulli and Genocchi Numbers

Let 𝑝 be a fixed odd prime number. Throughout this section, ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of ℚ_𝑝, respectively. Let 𝑣_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝑣𝑝(𝑝)}=1/𝑝. Let 𝑈𝐷(ℤ_𝑝) be the space of uniformly differentiable functions on ℤ_𝑝. For 𝑓∈𝑈𝐷(ℤ_𝑝), the bosonic 𝑝-adic integral on ℤ_𝑝 is defined by 𝐼𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇(𝑥)=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1𝑥=0}𝑓(𝑥) (3.1) (see [2, 5, 11]). From (3.1), we can derive the following integral equation: 𝐼𝑓_𝑛=𝐼𝑓+^𝑛−1_𝑖=0𝑓^′(𝑖)(𝑛∈ℕ), (3.2) where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛) and 𝑓′(𝑖)=((𝑑𝑓(𝑥))/𝑑𝑥)|_𝑥=𝑖 (see [2]). Let us take 𝑓(𝑦)=𝑒^{𝑡(𝑥+𝑦)}. Then we have _ℤ𝑝𝑒^{𝑡(𝑥+𝑦)}𝑦=𝑡𝑑𝜇𝑒^𝑡𝑒−1^𝑥𝑡=^∞_𝑛=0𝐵_𝑛(𝑥)𝑡^𝑛𝑛! (3.3) (see [2, 5]). From (3.3), we have _ℤ𝑝(𝑥+𝑛)^𝑛𝑑𝜇𝑦=𝐵_𝑛(𝑥),_ℤ𝑝𝑦^𝑛𝑑𝜇𝑦=𝐵_𝑛 (3.4) (see [2, 5]). Thus, by (3.2) and (3.4), we get _ℤ𝑝(𝑥+𝑛)^𝑚𝑑𝜇(𝑥)=_ℤ𝑝𝑥^𝑚𝑑𝜇(𝑥)+𝑚^𝑛−1_𝑖=0𝑖^𝑚−1, (3.5) (see [2]). From (3.5), we have 𝐵_𝑚(𝑛)−𝐵_𝑚=𝑚^𝑛−1_𝑖=0𝑖^𝑚−1𝑛∈ℤ₊ (3.6) (see [2]). The fermionic 𝑝-adic integral on ℤ_𝑝 is defined by Kim as follows [2, 8, 9]: 𝐼₋₁𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1}_𝑥=0𝑓(𝑥)(−1)^𝑥. (3.7) From (3.7), we obtain the following integral equation: 𝐼₋₁𝑓_𝑛=(−1)^𝑛𝐼₋₁𝑓+2^𝑛−1_𝑙=0(−1)^{𝑛−𝑙−1}𝑓(𝑙) (3.8) (see [2]), where 𝑓_𝑛(𝑥)=𝑓(𝑥+𝑛). Thus, by (3.8), we have _ℤ𝑝(𝑥+𝑛)^𝑚𝑑𝜇₋₁(𝑥)=(−1)^𝑛_ℤ𝑝𝑥^𝑚𝑑𝜇₋₁(𝑥)+2^𝑛−1_𝑙=0(−1)^{𝑛−𝑙−1}𝑙^𝑚 (3.9) (see [2]). Let us take 𝑓(𝑦)=𝑒^{𝑡(𝑥+𝑦)}. Then we have 𝑡_ℤ𝑝𝑒^{𝑡(𝑥+𝑦)}𝑑𝜇₋₁𝑦=2𝑡𝑒^𝑥𝑡𝑒^𝑡=+1^∞_𝑛=0𝐺_𝑛(𝑥)𝑡^𝑛.𝑛! (3.10) From (3.10), we have _ℤ𝑝𝑥+𝑦^𝑛𝑑𝜇₋₁𝑦=𝐺_𝑛+1(𝑥),𝑛+1_ℤ𝑝𝑦^𝑛𝑑𝜇₋₁𝑦=𝐺_𝑛+1.𝑛+1 (3.11) Thus, by (3.9) and (3.11), we get 𝐺_𝑚+1(𝑛)=()𝑚+1−1^𝑛𝐺_𝑛+1𝑛+1+2^𝑛−1_𝑙=0(−1)^𝑙−1𝑙^𝑚. (3.12) Let us consider the following 𝑝-adic integral on ℤ_𝑝: 𝐾₁=_ℤ𝑝𝐵_𝑛(𝑥)𝑑𝜇(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐵_{𝑛−𝑙ℤ𝑝}𝑥^𝑙𝑑𝜇(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝐵_𝑙. (3.13) From Theorem 2.1 and (3.13), one has 𝐾₁=^𝑛+1_{𝑘=0𝑘≠𝑛}𝑘𝑛+1𝐵_{𝑛+1−𝑘}𝑛+1^𝑘_𝑙=0𝑘𝑙𝐺_{𝑘−𝑙ℤ𝑝}𝑥^𝑙(𝑥)=𝑑𝜇_{𝑛+1𝑘𝑘=0𝑘≠𝑛𝑙=0}𝑘𝑘𝑙𝑛+1𝐵_{𝑛+1−𝑘}𝐵_𝑙𝐺_𝑘−𝑙.𝑛+1 (3.14) Therefore, by (3.13) and (3.14), we obtain the following theorem.

Theorem 3.1. For 𝑛∈ℤ₊, one has ^𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝐵_𝑙=^𝑛+1𝑘_{𝑘=0𝑘≠𝑛𝑙=0}𝑘𝑘𝑙𝑛+1𝐵_{𝑛+1−𝑘}𝐵_𝑙𝐺_𝑘−𝑙.𝑛+1 (3.15)

Now, one sets 𝐾₂=_ℤ𝑝𝐵_𝑛(𝑥)𝑑𝜇₋₁(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝐺_𝑙+1𝑙+1. (3.16) By Theorem 2.1, one gets 𝐾₂=^𝑛+1_{𝑘=0𝑘≠𝑛}𝑘𝑛+1𝐵_{𝑛+1−𝑘}𝑛+1^𝑘_𝑙=0𝑘𝑙𝐺_{𝑘−𝑙ℤ𝑝}𝑥^𝑙𝑑𝜇₋₁(𝑥)=^𝑛+1𝑘_{𝑘=0𝑘≠𝑛𝑙=0}𝑘𝑘𝑙𝑛+1𝐵_{𝑛+1−𝑘}𝐺_𝑘−𝑙𝐺_𝑙+1().𝑛+1)(𝑙+1 (3.17) Therefore, by (3.16) and (3.17), we obtain the following theorem.

Theorem 3.2. For 𝑛∈ℤ₊, one has ^𝑛_𝑙=0𝑛𝑙𝐵_𝑛−𝑙𝐺_𝑙+1=𝑙+1^𝑛+1𝑘_{𝑘=0𝑘≠𝑛𝑙=0}𝑘𝑘𝑙𝑛+1𝐵_{𝑛+1−𝑘}𝐺_𝑘−𝑙𝐺_𝑙+1().𝑛+1)(𝑙+1 (3.18)

Let us consider the following 𝑝-adic integral on ℤ_𝑝: 𝐾₃=_ℤ𝑝𝐺_𝑛(𝑥)𝑑𝜇₋₁(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐺_{𝑛−𝑙ℤ𝑝}𝑥^𝑙𝑑𝜇₋₁(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐺_𝑛−𝑙𝐺_𝑙+1𝑙+1. (3.19) From Theorem 2.2, one has 𝐾₃=−2^𝑛_𝑙=1𝑛𝑙𝐺_𝑙+1𝑙+1^𝑛−𝑙_𝑘=0𝑘𝐵𝑛−𝑙_{𝑛−𝑙−𝑘ℤ𝑝}𝑥^𝑘𝑑𝜇₋₁(𝑥)=−2^𝑛_𝑙=1^𝑛−𝑙_𝑘=0𝑛𝑙𝑘𝐵𝑛−𝑙_{𝑛−𝑙−𝑘}𝐺_𝑙+1𝐺_𝑘+1().𝑙+1)(𝑘+1 (3.20) Therefore, by (3.19) and (3.20), we obtain the following theorem.

Theorem 3.3. For 𝑛∈ℤ₊, one has ^𝑛_𝑙=0𝑛𝑙𝐺_𝑛−𝑙𝐺_𝑙+1𝑙+1=−2^𝑛_𝑙=1^𝑛−𝑙_𝑘=0𝑛𝑙𝑘𝑛−𝑙𝐵_{𝑛−𝑙−𝑘}𝐺_𝑙+1𝐺_𝑘+1(𝑙+1)(𝑘+1). (3.21)

Now, one sets 𝐾₄=_ℤ𝑝𝐺_𝑛(𝑥)𝑑𝜇(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐺_𝑛−𝑙𝐵_𝑙. (3.22) By Theorem 2.2, one gets 𝐾₄=−2^𝑛_𝑙=1^𝑛−𝑙_𝑘=0𝑛𝑙𝑘𝑛−𝑙𝐺_𝑙+1𝐵𝑙+1^{𝑛−𝑙−𝑘}𝐵_𝑘. (3.23) Therefore, by (3.22) and (3.23), we obtain the following corollary.

Corollary 3.4. For 𝑛∈ℤ₊, one has ^𝑛_𝑙=0𝑛𝑙𝐺_𝑛−𝑙𝐵_𝑙=−2^𝑛_𝑙=1^𝑛−𝑙_𝑘=0𝑛𝑙𝑘𝑛−𝑙𝐺_𝑙+1𝐵_{𝑛−𝑙−𝑘}𝐵_𝑘𝑙+1. (3.24)

Acknowledgement

This research was supported by Kyungpook National University research Fund, 2012.