A Note on Eulerian Polynomials

Abstract

We study Genocchi, Euler, and tangent numbers. From those numbers we derive some identities on Eulerian polynomials in connection with Genocchi and tangent numbers.

1. Introduction

As is well known, the Eulerian polynomials, 𝐴_𝑛(𝑡), are defined by generating function as follows: 1−𝑡exp(𝑥(𝑡−1))−𝑡=𝑒^{𝐴(𝑡)𝑥}=^∞_𝑛=0𝐴_𝑛(𝑡)𝑥^𝑛,𝑛! (1.1) with the usual convention about replacing 𝐴^𝑛(𝑡) by 𝐴_𝑛(𝑡) (see [1–18]). From (1.1), we note that (𝐴(𝑡)+(𝑡−1))^𝑛−𝑡𝐴_𝑛(𝑡)=(1−𝑡)𝛿_0,𝑛, (1.2) where 𝛿_𝑛,𝑘 is the Kronecker symbol (see [3]).

Thus, by (1.2), we get 𝐴₀(𝑡)=1,𝐴_𝑛(𝑡)=1𝑡−1^𝑛−1_𝑙=0𝑛𝑙𝐴_𝑙(𝑡)(𝑡−1)^𝑛−𝑙,(𝑛≥1). (1.3)

By (1.1), (1.2), and (1.3), we see that ^𝑚_𝑖=1𝑖^𝑛𝑡^𝑖=^𝑛_𝑙=1(−1)^𝑛+𝑙𝑛𝑙𝑡^𝑚+1𝐴_𝑛−𝑙(𝑡)(𝑡−1)^{𝑛−𝑙+1}𝑚^𝑙+(−1)^𝑛𝑡𝑡^𝑚−1(𝑡−1)^𝑛+1𝐴_𝑛(𝑡), (1.4) where 𝑚≥1 and 𝑛≥0 (see [1]).

The Genocchi polynomials are defined by 2𝑡𝑒^𝑡𝑒+1^𝑥𝑡=𝑒^{𝐺(𝑥)𝑡}=^∞_𝑛=0𝐺_𝑛(𝑥)𝑡^𝑛,𝑛! (1.5)(see [6–18]). In the special case, 𝑥=0, 𝐺_𝑛(0)=𝐺_𝑛 are called the 𝑛th Genocchi numbers (see [14, 17, 18]).

It is well known that the Euler polynomials are also defined by 2𝑒^𝑡𝑒+1^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛,𝑛! (1.6)(see [1–5, 19–24]). Here 𝑥=0, then 𝐸_𝑛(0)=𝐸_𝑛 is called the 𝑛th Euler number. From (1.6), we have 𝐸₀=1,(𝐸+1)^𝑛+𝐸_𝑛=2𝛿_0,𝑛, (1.7)(see [3–5, 19–23]).

As is well known, the Bernoulli numbers are defined by 𝐵₀=1,(𝐵+1)^𝑛−𝐵_𝑛=𝛿_0,𝑛, (1.8)(see [5, 18, 19]), with the usual convention about replacing 𝐵^𝑛 by 𝐵_𝑛.

From (1.8), we note that the Bernoulli polynomials are also defined as 𝐵_𝑛(𝑥)=^𝑛_𝑙=0𝑛𝑙𝐵_𝑙𝑥^𝑛−𝑙=(𝐵+𝑥)^𝑛, (1.9)(see [5, 18, 19]).

The tangent numbers 𝑇_2𝑛−1(𝑛≥1) are defined as the coefficients of the Taylor expansion of tan𝑥: tan𝑥=^∞_𝑛=1𝑇_2𝑛−1()!𝑥2𝑛−1^2𝑛−1=𝑥+𝑥1!³𝑥3!2+⁵5!16+⋯, (1.10)(see [1–3, 5]).

In this paper, we give some identities on the Eulerian polynomials at 𝑡=−1 associated with Genocchi, Euler, and tangent numbers.

2. Witt's Formula for Eulerian Polynomials

In this section, we assume that ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will, respectively, denote the ring of 𝑝-adic integers, the field of 𝑝-adic numbers, and the completion of algebraic closure of ℚ_𝑝. The 𝑝-adic norm is normalized so that |𝑝|_𝑝=1/𝑝.

Let 𝑞 be an indeterminate with |1−𝑞|_𝑝<1. Then the 𝑞-number is defined by [𝑥]_𝑞=1−𝑞^𝑥,[𝑥]1−𝑞^−𝑞=1−−𝑞^𝑥,1+𝑞 (2.1)(see [6–18]).

Let 𝐶(ℤ_𝑝) be the space of continuous functions on ℤ_𝑝. For 𝑓∈𝐶(ℤ_𝑝), the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by 𝐼_−𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)−𝑞^𝑥, (2.2) (see [7, 10–13]). From (2.2), we can derive the following: 𝑞⁻¹𝐼_−𝑞−1𝑓₁+𝐼_−𝑞−1𝑓=[2]_𝑞−1𝑓(0), (2.3) where 𝑓₁(𝑥)=𝑓(𝑥+1).

Let us take 𝑓(𝑥)=𝑒^{−𝑥(1+𝑞)𝑡}. Then, by (2.3), we get 𝑞+𝑒^{−(1+𝑞)𝑡}𝑞_ℤ𝑝𝑒^{−𝑥(1+𝑞)𝑡}𝑑𝜇_−𝑞−1(𝑥)=[2]_𝑞−1. (2.4)

Thus, from (2.4), we have _ℤ𝑝𝑒^{−𝑥(1+𝑞)𝑡}𝑑𝜇_−𝑞−1(𝑥)=1+𝑞𝑒^{−(1+𝑞)𝑡}=+𝑞^∞_𝑛=0𝐴_𝑛𝑡−𝑞^𝑛.𝑛! (2.5)

By Taylor expansion on the left-hand side of (2.5), we get ^∞_𝑛=0(−1)^𝑛_ℤ𝑝𝑥^𝑛𝑑𝜇_−𝑞−1(𝑥)1+𝑞^𝑛𝑡^𝑛=𝑛!^∞_𝑛=0𝐴_𝑛𝑡−𝑞^𝑛.𝑛! (2.6)

Comparing coefficients on the both sides of (2.6), we have _ℤ𝑝𝑥^𝑛𝑑𝜇_−𝑞−1(𝑥)=(−1)^𝑛1+𝑞^𝑛𝐴_𝑛−𝑞. (2.7) Therefore, by (2.7), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℤ₊, one has _ℤ𝑝x^𝑛𝑑𝜇_−𝑞−1(𝑥)=(−1)^𝑛1+𝑞^𝑛𝐴_𝑛−𝑞, (2.8) where 𝐴_𝑛(−𝑞) is an Eulerian polynomials.

It seems interesting to study Theorem 2.1 at 𝑞=1. By (2.3), we get 𝐼₋₁𝑓₁+𝐼₋₁𝑓=2𝑓(0), (2.9) where 𝑓₁(𝑥)=𝑓(𝑥+1). From (2.9), we can derive the following equation: _ℤ𝑝𝑓(𝑥+𝑛)𝑑𝜇₋₁(𝑥)+(−1)^𝑛−1_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=2^𝑛−1_𝑙=0(−1)^{𝑛−𝑙+1}𝑓(𝑙), (2.10) where 𝑛∈ℤ₊ (see [5–13]).

From (2.9), we can derive the following: 0=_ℤ𝑝sin𝑎(𝑥+1)𝑑𝜇₋₁(𝑥)+_ℤ𝑝sin𝑎𝑥𝑑𝜇₋₁(𝑥)=(cos𝑎+1)_ℤ𝑝sin𝑎𝑥𝑑𝜇₋₁(𝑥)+sin𝑎_ℤ𝑝cos𝑎𝑥𝑑𝜇₋₁(𝑥),2=_ℤ𝑝cos𝑎(𝑥+1)𝑑𝜇₋₁(𝑥)+_ℤ𝑝cos𝑎𝑥𝑑𝜇₋₁(𝑥)=(cos𝑎+1)_ℤ𝑝cos𝑎𝑥𝑑𝜇₋₁(𝑥)−sin𝑎_ℤ𝑝sin𝑎𝑥𝑑𝜇₋₁(𝑥). (2.11) By (2.11), we get _ℤ𝑝sin𝑎𝑥𝑑𝜇₋₁(𝑥)=−sin𝑎𝑎cos𝑎+1=−tan2. (2.12) From (1.10) and (2.12), we have ^∞_𝑛=1𝑇_2𝑛−1()!𝑎2𝑛−12^2𝑛−1=−_ℤ𝑝sin𝑎𝑥𝑑𝜇₋₁(𝑥)=^∞_𝑛=1(−1)^𝑛𝑎^2𝑛−1(2𝑛−1)!_ℤ𝑝𝑥^2𝑛−1𝑑𝜇₋₁(𝑥). (2.13) By comparing coefficients on the both sides of (2.13), we get _ℤ𝑝𝑥^2𝑛−1𝑑𝜇₋₁(𝑥)=(−1)^𝑛𝑇_2𝑛−12^2𝑛−1,for𝑛∈ℕ, (2.14) where 𝑇_2𝑛−1 is the (2𝑛−1)th tangent number.

Therefore, by (2.14), we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℕ, one has _ℤ𝑝𝑥^2𝑛−1𝑑𝜇₋₁(𝑥)=(−1)^𝑛𝑇_2𝑛−12^2𝑛−1, (2.15) where 𝑇_2𝑛−1 is the (2𝑛−1)th tangent numbers.

From Theorem 2.1, one has _ℤ𝑝𝑥^𝑛𝑑𝜇₋₁(𝑥)=(−1)^𝑛2^𝑛𝐴_𝑛(−1). (2.16) Therefore, by Theorem 2.2 and (2.16), we obtain the following corollary.

Corollary 2.3. For 𝑛∈ℕ, one has 𝐴_2𝑛−1(−1)=(−1)^𝑛−1𝑇_2𝑛−1. (2.17)

From (1.6) and (2.9), we have _ℤ𝑝𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)=2𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛𝑡^𝑛,𝑛! (2.18) (see [5]). Thus, by (2.16) and (2.18), we get _ℤ𝑝𝑥^2𝑛−1𝑑𝜇₋₁(𝑥)=𝐸_2𝑛−1=(−1)^𝑛𝑇_2𝑛−12^2𝑛−1. (2.19) Therefore, by Corollary 2.3 and (2.19), we obtain the following corollary.

Corollary 2.4. For 𝑛∈ℕ, one has 𝐸_2𝑛−1=(−1)^𝑛𝑇_2𝑛−12^2𝑛−1𝐴=−^2𝑛−1(−1)2^2𝑛−1. (2.20)

By (1.5) and (2.9), we get 𝑡_ℤ𝑝𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)=2𝑡𝑒^2𝑡𝑒−1^𝑡−2𝑡𝑒^2𝑡=−1_∞𝑛=0𝐵_𝑛122^𝑛𝑡^𝑛−𝑛!^∞_𝑛=02^𝑛𝐵_𝑛𝑡𝑛!^𝑛=^∞_𝑛=0𝐵_𝑛12−𝐵_𝑛2^𝑛𝑡^𝑛.𝑛! (2.21)

By (2.21), we get _ℤ𝑝𝑥^𝑛𝑑𝜇₋₁(𝑥)=𝐵_𝑛+1(1/2)−𝐵_𝑛+12𝑛+1^𝑛+1. (2.22)

Thus, from (2.19), Theorem 2.2 and Corollary 2.3, we have 𝐵_2𝑛(1/2)−𝐵_2𝑛2^2𝑛=()2𝑛−1^𝑛𝑇_2𝑛−12^2𝑛−1𝐴=−^2𝑛−1(−1)2^2𝑛−1. (2.23)

Therefore, by (2.23), we obtain the following theorem.

Theorem 2.5. For 𝑛∈ℕ, one has 𝐵_2𝑛(1/2)−𝐵_2𝑛2^2𝑛𝑛=(−1)^𝑛𝑇_2𝑛−12^2𝑛−2𝐴=−^2𝑛−1(−1)2^2𝑛−2. (2.24)

From (1.5), we note that 𝑡_ℤ𝑝𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)=2𝑡𝑒^𝑡=+1^∞_𝑛=0𝐺_𝑛𝑡^𝑛𝑛! (2.25) (see [13, 14]). Thus, by (2.25), we get 𝐺₀=0,(𝐺+1)^𝑛+𝐺_𝑛=2𝛿_1,𝑛, (2.26) (see [13, 14]), with the usual convention about replacing 𝐺^𝑛 by 𝐺_𝑛.

From (1.5) and (2.9), one has 𝑡_ℤ𝑝𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)𝑡=2𝑒^𝑡−−12𝑡𝑒^2𝑡−1=2^∞_𝑛=0𝐵_𝑛−2^𝑛𝐵_𝑛𝑡^𝑛.𝑛! (2.27) Thus, by (2.27), we get _ℤ𝑝𝑥^𝑛𝑑𝜇₋₁(𝑥)𝐵=2^𝑛+1−2^𝑛+1𝐵_𝑛+1.𝑛+1 (2.28)

From (2.28), we have 𝐺_2𝑛=2𝑛_ℤ𝑝𝑥^2𝑛−1𝑑𝜇₋₁(𝑥)=𝐵_2𝑛−2^2𝑛𝐵_2𝑛𝑛,for∈ℕ. (2.29) Therefore, by (2.19), Corollary 2.3 and (2.29), we obtain the following theorem.

Theorem 2.6. For 𝑛∈ℕ, we have 𝐺_2𝑛=2𝐵_2𝑛−2^2𝑛𝐵_2𝑛. (2.30)
In particular, −12^2𝑛−1𝐴_2𝑛−1(−1)=(−1)^𝑛𝑇_2𝑛−112^2𝑛−1=𝐺_2𝑛.2𝑛 (2.31)

3. Further Remark

In complex plane, we note that 1tan𝑥=𝑖𝑒^𝑖𝑥−𝑒^−𝑖𝑥𝑒^𝑖𝑥+𝑒^−𝑖𝑥=1𝑖1−2𝑒^−𝑖𝑥𝑒^𝑖𝑥+𝑒^−𝑖𝑥=1𝑖1−^∞_𝑛=0𝐸_𝑛2𝑛!^𝑛𝑖^𝑛𝑥^𝑛=1𝑖−^∞_n=1𝐸_𝑛2𝑛!^𝑛𝑖^𝑛𝑥^𝑛=^∞_𝑛=1(−1)^𝑛()!𝐸2𝑛−1^2𝑛−12^2𝑛−1𝑥^2𝑛−1. (3.1) By (1.10) and (3.1), we also get 𝑇_2𝑛−1=(−1)^𝑛𝐸_2𝑛−12^2𝑛−1,for𝑛∈ℕ. (3.2) From (1.5), we have ^∞_𝑛=1𝑡^2𝑛()!𝐺2𝑛^2𝑛=^∞_𝑛=1(𝑖𝑡)^2𝑛()!()2𝑛−1^𝑛𝐺_2𝑛=2𝑖𝑡1+𝑒^𝑖𝑡=−𝑖𝑡𝑖𝑡1−𝑒^𝑖𝑡1+𝑒^𝑖𝑡=𝑖𝑡𝑒^{−𝑖𝑡/2}−𝑒^𝑖𝑡/2𝑒^𝑖𝑡/2+𝑒^{−𝑖𝑡/2}𝑒=𝑡^𝑖𝑡/2−𝑒^{−𝑖𝑡/2}/2𝑖𝑒^𝑖𝑡/2+𝑒^{−𝑖𝑡/2}𝑡/2=𝑡tan2. (3.3) Thus, by (1.10) and (3.3), we get ^∞_𝑛=1𝑡^2𝑛()!𝐺2𝑛^2𝑛𝑡=𝑡tan2=𝑡^∞_𝑛=1(𝑡/2)^2𝑛−1()!𝑇2𝑛−1^2𝑛−1=^∞_𝑛=1𝑡^2𝑛(2𝑛−1)!2^2𝑛−1𝑇_2𝑛−1. (3.4)

From (3.4), we have 𝑛𝑇_2𝑛−1=2^2𝑛−2𝐺_2𝑛=2^2𝑛−11−2^2𝑛𝐵_2𝑛. (3.5) By (1.1), we see that 21+𝑒^−2𝑖𝑡=^∞_𝑛=0𝐴_𝑛()𝑖−1^𝑛𝑡^𝑛.𝑛! (3.6) Thus, we note that ^∞_𝑛=1𝑖^𝑛−1𝐴_𝑛()𝑡−1^𝑛=1𝑛!𝑖21+𝑒^−2𝑖𝑡−1=1−𝑒^−2𝑖𝑡1+𝑒^−2𝑖𝑡𝑖=𝑒^𝑖𝑡−𝑒^−𝑖𝑡/2𝑒^𝑖𝑡+𝑒^−𝑖𝑡/2𝑖=tan𝑡=^∞_𝑛=1𝑇_2𝑛−1𝑡^2𝑛−1()!.2𝑛−1 (3.7)

From (3.7), we have 𝐴_2𝑛(−1)=0,𝐴_2𝑛−1(−1)=(−1)^𝑛−1𝑇_2𝑛−1,(𝑛≥1). (3.8) It is easy to show that ^𝑚_𝑘=1𝑘^𝑛(−1)^𝑘=(−1)^𝑚𝑛_𝑘=0𝑛𝑘𝐴_𝑘(−1)2^𝑘+1𝑚^𝑛−𝑘−(−1)^𝑚−12^𝑛+1𝐴_𝑛(−1). (3.9) For simple calculation, we can derive the following equation: 𝑒𝑖tan𝑥=^𝑖𝑥−𝑒^−𝑖𝑥𝑒^𝑖𝑥+𝑒^−𝑖𝑥2=1−𝑒^2𝑖𝑥+4−1𝑒^4𝑖𝑥.−1 (3.10) By (3.10), we get 𝑥tan𝑥=−𝑖𝑥+2𝑖𝑥𝑒^2𝑖𝑥−−14𝑖𝑥𝑒^4𝑖𝑥=−1^∞_𝑛=1(−1)^𝑛𝐵_2𝑛4^𝑛1−4^𝑛()!𝑥2𝑛^2𝑛. (3.11) Thus, from (3.11),we have tan𝑥=^∞_𝑛=1(−1)^𝑛𝐵_2𝑛4^𝑛1−4^𝑛()!𝑥2𝑛^2𝑛−1. (3.12)

By (1.10) and (3.12), we get 𝑇_2𝑛−1=(−1)^𝑛𝐵_2𝑛4^𝑛1−4^𝑛2𝑛,for𝑛∈ℕ. (3.13) From Corollary 2.3 and (3.13), we can derive the following identity: 𝐴_2𝑛−1()𝐵−1=−^2𝑛2^2𝑛−11−4^𝑛𝑛. (3.14)

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 2012R1A1A2003786. Also, the authors would like to thank the referees for their valuable comments and suggestions.