Some Identities on the High-Order 𝑞-Euler Numbers and Polynomials with Weight 0

Abstract

We construct the 𝑁th order nonlinear ordinary differential equation related to the generating function of 𝑞-Euler numbers with weight 0. From this, we derive some identities on 𝑞-Euler numbers and polynomials of higher order with weight 0.

1. Introduction

As a well-known definition, the Euler polynomial 𝐸_𝑛(𝑥) is given by 2𝑒^𝑡𝑒+1^𝑥𝑡=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛.𝑛! (1) In the special case, 𝑥=0, 𝐸_𝑛(0)=𝐸_𝑛 is the 𝑛th Euler number.

From (1), we note that 𝐸₀=1,(𝐸+1)^𝑛+𝐸_𝑛=0,if𝑛>0, (2) with the usual convention of replacing 𝐸^𝑛 by 𝐸_𝑛 (see [1–16]).

In the viewpoint of the 𝑞-extension of (1) and (2), let us consider the following 𝑞-Euler number and polynomial: 2𝑞𝑒^𝑡𝑒+1^𝑥𝑡=^∞∼_𝑛=0𝐸_𝑛,𝑞(𝑥)𝑡^𝑛,𝑛!_∼𝐸_0,𝑞=21+𝑞,𝑞𝐸^∼_𝑞+1^𝑛+𝐸^∼_𝑛,𝑞=0,if𝑛>0, (3)(4) with the usual convention of replacing 𝐸^∼𝑛_𝑞 by 𝐸^∼_𝑛,𝑞.

Equation (3) is called the generating function of 𝑞-Euler polynomial with weight 0. In the case 𝑥=0, 𝐸^∼_𝑛,𝑞(0)=𝐸^∼_𝑛,𝑞 is the 𝑛th 𝑞-Euler number with weight 0 (see [5, 11]).

Throughout this paper, let 𝑞 be a complex number with |𝑞|<1. As 𝑞→1, we obtain (1) and (2) from (3) and (4).

The generating function of Eulerian polynomial 𝐻_𝑛(𝑥∣𝑢) is defined by 1−𝑢𝑒^𝑡𝑒−𝑢^𝑥𝑡=^∞_𝑛=0𝐻_𝑛()𝑡𝑥∣𝑢^𝑛,𝑛! (5) where 𝑢∈ℂ with 𝑢≠1. In the special case, 𝑥=0, 𝐻_𝑛(0∣𝑢)=𝐻_𝑛(𝑢) is called the 𝑛th Eulerian number (see [1–3]). Sometimes that is called the 𝑛th Frobenius-Euler number (see [9–11, 15]).

From (1) and (5), we note that 𝐻_𝑛(𝑥∣−1)=𝐸_𝑛(𝑥). From (5), we have 𝐻₀(𝑢)=1,𝐻_𝑛(1∣𝑢)−𝑢𝐻_𝑛(𝑢)=(1−𝑢)𝛿_0,𝑛, (6) where 𝛿_𝑛,𝑘 is Kronecker symbol (see [9–11]).

For 𝑁∈ℕ, the 𝑞-Euler polynomial of order 𝑁 is defined by the generating function as follows: 𝐺^𝑁_𝑞()=2𝑡,𝑥𝑞𝑒^𝑡2+1×⋯×𝑞𝑒^𝑡+1_𝑁-times𝑒^𝑥𝑡=^∞∼_𝑛=0𝐸^(𝑁)_𝑛,𝑞(𝑥)𝑡^𝑛.𝑛! (7) In the special case, 𝑥=0, 𝐸^∼(𝑁)_𝑛,𝑞(0)=𝐸^∼(𝑁)_𝑛,𝑞 is called the 𝑛th 𝑞-Euler number of order 𝑁 with weight 0 (see [5, 11]).

In [9], Kim derived some identities between the sums of products of Frobenius-Euler polynomials and Frobenius-Euler polynomials of higher order. The main idea is to construct nonlinear ordinary differential equations with respect to 𝑡 which are closely related to the generating function of Frobenius-Euler polynomial. In [3], Choi considered nonlinear ordinary differential equations with respect to 𝑢 not 𝑡.

In this paper, we construct nonlinear ordinary differential equations with respect to 𝑞. The purpose of this paper is to give some new identities on the high order 𝑞-Euler numbers and polynomials with weight 0 by using the differential equations of 𝑞.

2. Construction of Nonlinear Differential Equations

We define 𝑞=1𝐺=𝐺𝑞𝑒^𝑡,𝐺+1_𝑁(𝑡,𝑥)=𝐺×⋯×𝐺_𝑁-times𝑒^𝑥𝑡for𝑁∈ℕ. (8) From (7) and (8), we note that 𝐺^𝑁_𝑞(𝑡,𝑥)=2^𝑁𝐺^𝑁(𝑡,𝑥)=2^𝑁𝐺^𝑁𝑒^𝑥𝑡. (9) By differentiating (8) with respect to 𝑞, we get 𝐺⁽¹⁾=𝑑𝐺𝑑𝑞=−𝑞𝑒^𝑡+1−1𝑞𝑞𝑒^𝑡+1²𝐺=−𝑞+𝐺²𝑞,𝑞𝐺⁽¹⁾+𝐺=𝐺². (10) By differentiating (10) with respect to 𝑞, we get 𝑞²𝐺⁽²⁾+4𝑞𝐺⁽¹⁾+2𝐺=2!𝐺³, (11) where 𝐺^(𝑁)=𝑑^𝑁𝐺/𝑑𝑞^𝑁.

By the derivative of (11) with respect to 𝑞, we have 𝑞³𝐺⁽³⁾+9𝑞²𝐺⁽²⁾+18𝑞𝐺⁽¹⁾+3!𝐺=3!𝐺⁴. (12) Continuing this process, we get (𝑁−1)!𝐺^𝑁=^𝑁−1_𝑘=0𝑎_𝑘(𝑁)𝑞^𝑘𝐺^(𝑘). (13)

Let us consider the derivative of (13) with respect to 𝑞 to find the coefficient 𝑎_𝑘(𝑁) in (13).

By (10), we get 𝑞𝑑)𝑑𝑞(𝑁−1!𝐺^𝑁=𝑁!𝐺^𝑁−1𝑞𝐺⁽¹⁾=𝑁!𝐺^𝑁−1−𝐺+𝐺²=𝑁!𝐺^𝑁+1−𝑁(𝑁−1)!𝐺^𝑁. (14) From (13) and (14), we get 𝑁!𝐺^𝑁+1=𝑁(𝑁−1)!𝐺^𝑁+^𝑁−1_𝑘=0𝑘𝑎_𝑘(𝑁)𝑞^𝑘𝐺^(𝑘)+^𝑁_𝑘=1𝑎_𝑘−1(𝑁)𝑞^𝑘𝐺^(𝑘), (15) where 𝑁!𝐺^𝑁+1=∑^𝑁_𝑘=0𝑎_𝑘(𝑁+1)𝑞^𝑘𝐺^(𝑘).

By comparing coefficients on both sides of (15), we obtain the following recurrence relations: 𝑎₀(𝑁+1)=𝑁𝑎₀(𝑁),𝑎_𝑁(𝑁+1)=𝑎_𝑁−1(𝑁),𝑎_𝑘(𝑁+1)=𝑁𝑎_𝑘(𝑁)+𝑘𝑎_𝑘(𝑁)+𝑎_𝑘−1(𝑁), (16)(17) for 1≤𝑘≤𝑁−1 and 𝑎_𝑘(𝑁)=0.

From the first part of (16), we have 𝑎₀(𝑁+1)=𝑁𝑎₀(𝑁)=𝑁(𝑁−1)𝑎₀(𝑁−1)=⋯=𝑁!𝑎₀(2). (18) By (10) and (13), we have 𝑞𝐺⁽¹⁾+𝐺=𝐺²=¹_𝑘=0𝑎_𝑘(2)𝑞^𝑘𝐺^(𝑘)=𝑎₀(2)𝐺+𝑎₁(2)𝑞𝐺⁽¹⁾. (19) From (18) and (19), we get 𝑎₀(2)=1,𝑎₁(2)=1,𝑎₀(𝑁)=(𝑁−1)!. (20) From the second part of (16), we have 𝑎_𝑁(𝑁+1)=𝑎_𝑁−1(𝑁)=⋯=𝑎₁(2)=1. (21)

To find 𝑎_𝑘(𝑁) in (13) from (17), we set 𝑔(𝑡,𝑠)=_{𝑁≥10≤𝑘≤𝑁−1}𝑎_𝑘(𝑁)𝑡^𝑁𝑠𝑁!^𝑘, (22) where |𝑡|<1 (see [9]).

From (17) and (22), we have _{𝑁≥10≤𝑘≤𝑁−1}𝑎_𝑘+1()𝑡𝑁+1^𝑁𝑠𝑁!^𝑘=_{𝑁≥10≤𝑘≤𝑁−1}𝑁𝑎_𝑘−1(𝑁)𝑡^𝑁𝑠𝑁!^𝑘+_{𝑁≥10≤𝑘≤𝑁−1}(𝑘+1)𝑎_𝑘+1(𝑁)𝑡^𝑁𝑠𝑁!^𝑘+𝑔(𝑡,𝑠). (23) From the left hand side of (23), we have _{𝑁≥10≤𝑘≤𝑁−1}𝑎_𝑘+1()𝑡𝑁+1^𝑁𝑠𝑁!^𝑘=1𝑠_{𝑁≥21≤𝑘≤𝑁−1}𝑎_𝑘(𝑁)𝑡^𝑁−1()!𝑠𝑁−1^𝑘=1𝑠_{𝑁≥20≤𝑘≤𝑁−1}𝑎_𝑘(𝑁)𝑡^𝑁−1()!𝑠𝑁−1^𝑘−𝑎₀(𝑁)𝑡^𝑁−1()!=1𝑁−1𝑠_{𝑁≥10≤𝑘≤𝑁−1}𝑎_𝑘(𝑁)𝑡^𝑁−1()!𝑠𝑁−1^𝑘−𝑎₀(1)−_𝑁≥2𝑡^𝑁−1=1𝑠𝑔_𝑡+1,𝑡−1 (24) where 𝑔_𝑡=𝜕𝑔/𝜕𝑡. From the first term of the right hand side of (23), we have _{𝑁≥10≤𝑘≤𝑁−1}𝑁𝑎_𝑘+1(𝑁)𝑡^𝑁𝑠𝑁!^𝑘=𝑡𝑠_{𝑁≥11≤𝑘≤𝑁−1}𝑎_𝑘(𝑁)𝑡^𝑁−1()!𝑠𝑁−1^𝑘=𝑡𝑠_{𝑁≥10≤𝑘≤𝑁−1}𝑎_𝑘(𝑁)𝑡^𝑁−1()!𝑠𝑁−1^𝑘−_𝑁≥1𝑎₀(𝑁)()!𝑡𝑁−1^𝑁−1=𝑡𝑠𝑔_𝑡+1.𝑡−1 (25) From the second term of the right hand side of (23), we have _{𝑁≥10≤𝑘≤𝑁−1}(𝑘+1)𝑎_𝑘+1(𝑁)𝑡^𝑁𝑠𝑁!^𝑘=_{𝑁≥11≤𝑘≤𝑁}𝑘𝑎_𝑘(𝑁)𝑡^𝑁𝑠𝑁!^𝑘−1=_{𝑁≥10≤𝑘≤𝑁−1}𝑘𝑎_𝑘(𝑁)𝑡^𝑁𝑠𝑁!^𝑘−1=𝑔_𝑠, (26) where 𝑔_𝑠=𝜕𝑔/𝜕𝑠.

From (22)–(26), we obtain the following initial value problem quasilinear first-order partial differential equation: (𝑡−1)𝑔_𝑡+𝑠𝑔_𝑠|𝑡|𝑔()=−𝑠𝑔−1,<1,0,𝑠=0,𝑠∈ℝ. (27)

We consider Cauchy problem for the following first-order quasilinear partial differential equation: 𝑃𝑥,𝑦,𝑧𝑧_𝑥+𝑄𝑥,𝑦,𝑧𝑧_𝑦,𝑧𝑥=𝑅𝑥,𝑦,𝑧₀(𝑡),𝑦₀(𝑡)=𝑧₀(𝑡),𝑡∈𝐼, (28) where 𝐼 is some interval.

We know that (28) has a unique solution under some conditions as follows.

Theorem A (see [17, page 65]). Suppose that 𝑃,𝑄, and 𝑅 are of class 𝐶¹ in a domain Ω of ℝ³ containing the point (𝑥₀,𝑦₀,𝑧₀) and suppose that 𝑃𝑥₀,𝑦₀,𝑧₀𝑑𝑦₀𝑡₀𝑥𝑑𝑡−𝑄₀,𝑦₀,𝑧₀𝑑𝑥₀𝑡₀𝑑𝑡≠0. (29) Then in a neighborhood 𝑈 of (𝑥₀,𝑦₀) there exists a unique solution of (28) at every point of initial curve contained in 𝑈.

Since (27) satisfies (29) and regularity conditions, there exists a unique solution of (27).

It is customary to write (27) in the form 𝑑𝑡=𝑡−1𝑑𝑠𝑠=𝑑𝑔,−𝑠𝑔−1𝑡=0,𝑠=𝑝,𝑔=0. (30)(31) Since 𝑑𝑡/(𝑡−1)=𝑑𝑠/𝑠 is separable, we get 𝑢₁𝑡,𝑠,𝑔=1−𝑡𝑠. (32)𝑢₁ is a solution of partial differential equation of (27).

From (30), we get the linear equation 𝑑𝑔1𝑑𝑠=−𝑔−𝑠. (33) By the integrating factor method, we have 𝑢₂𝑡,𝑠,𝑔=𝑒^𝑠𝑔+𝐸_𝑖(𝑠). (34) The exponential integral 𝐸_𝑖(𝑠) is defined by 𝐸_𝑖(𝑠)=^𝑠_−∞𝑒^𝑟𝑟|𝑠|+𝑑𝑟=𝛾+ln^∞_𝑛=1𝑠^𝑛,(),𝑛⋅𝑛!𝑠∈ℝ,𝑠≠0 (35) where 𝛾 is Euler constant.

𝑢₂ is another solution of partial differential equation of (27), and 𝑢₁ and 𝑢₂ are linearly independent.

From the parameterized initial conditions (31), (33), and (34), we get 𝑢₂=𝐸_𝑖1𝑢₁,𝑒^𝑠𝑔+𝐸_𝑖(𝑥)=𝐸_𝑖𝑠.1−𝑡 (36) Thus, from (35) and (36), we obtain the following unique solution of (27): 𝑔(𝑡,𝑠)=𝑒^−𝑠−ln|1−𝑡|+^∞_𝑛=1𝑠^𝑛1𝑛⋅𝑛!1−𝑡^𝑛−1. (37) Moreover, if we choose another initial condition 𝑔(𝑡,0)=^∞_𝑁≥1𝑎₀(𝑁)𝑡^𝑁=𝑁!^∞_𝑁≥1𝑡^𝑁𝑁 (38) from (20) and (22), then (37) satisfies it.

We note that 11−𝑡^𝑛−1=_𝑙1≥0𝑡^𝑙1×⋯×_𝑙𝑛≥0𝑡^𝑙𝑛_𝑛-times−1=_{𝑁≥1𝑙1+⋯+𝑙𝑛=𝑁}𝑡^𝑁=_𝑁≥1𝑁𝑡𝑛+𝑁−1_𝑁. (39) By (37) and (39), we get 𝑔(𝑡,𝑠)=_𝑘≥0(−1)^𝑘𝑠𝑘!^𝑘_𝑁≥1𝑡^𝑁𝑁+_𝑘≥0(−1)^𝑘𝑠𝑘!^𝑘×_𝑛≥1𝑠^𝑛𝑛⋅𝑛!_𝑁≥1𝑁𝑡𝑛+𝑁−1_𝑁=_{𝑁≥10≤𝑘≤𝑁−1}(−1)^𝑘𝑡𝑁⋅𝑘!^𝑁𝑠^𝑘+_{𝑁≥1𝑘≥𝑁}(−1)^𝑘𝑡𝑁⋅𝑘!^𝑁𝑠^𝑘+_{𝑁≥11≤𝑘≤𝑁−1}^𝑘_𝑙=1(−1)^𝑘−𝑙()𝑁𝑡𝑘−𝑙!𝑙⋅𝑙!𝑙+𝑁−1_𝑁𝑠^𝑘+_{𝑁≥1𝑘≥𝑁}^𝑘_𝑙=1(−1)^𝑘−𝑙()𝑁𝑡𝑘−𝑙!𝑙⋅𝑙!𝑙+𝑁−1_𝑁𝑠^𝑘. (40) It is known that (−1)^𝑙𝑁=𝑙+𝑁−1𝑙𝑁𝑙,−𝑁_𝑘𝑙=0𝑘𝑙𝑁𝑙=𝑘.𝑘+𝑁 (41) In the case of 𝑘≥𝑁 in (40), from (41), we get (−1)^𝑘+𝑁⋅𝑘!^𝑘_𝑙=1(−1)^𝑘−𝑙()𝑁=()𝑘−𝑙!𝑙⋅𝑙!𝑙+𝑁−1−1^𝑘1+1𝑁⋅𝑘!𝑁⋅𝑘!^𝑘_𝑙=1𝑘𝑙𝑙=()−𝑁−1^𝑘1𝑘𝑁⋅𝑘!1+𝑘−𝑁−1=0. (42) By (40) and (41), we get 𝑔(𝑡,𝑠)=_{𝑁≥11≤𝑘≤𝑁−1}(−1)^𝑘(𝑁−1)!𝑘𝑡𝑘!𝑘−𝑁^𝑁𝑠𝑁!^𝑘+_𝑁≥1()!𝑡𝑁−1^𝑁𝑁!=_{𝑁≥10≤𝑘≤𝑁−1}()!𝑘𝑁−𝑘−1𝑁−1²𝑡^𝑁𝑠𝑁!^𝑘, (43) where _𝑘^𝑘−𝑁=(−1)^𝑘_𝑘^𝑁−1. Thus, by (22) and (43), we get 𝑎_𝑘(𝑁)=()!𝑘𝑁−𝑘−1𝑁−1². (44) Therefore, by (13) and (44), we obtain the following theorem.

Theorem 1. For 𝑞∈ℂ with |𝑞|<1 and 𝑁∈ℕ, one can consider the following nonlinear (𝑁−1)th order ordinary differential equation with respect to 𝑞: 𝐺^𝑁𝑞=1(𝑁−1)!^𝑁−1_𝑘=0()!𝑘𝑁−𝑘−1𝑁−1²𝑞^𝑘𝐺^(𝑘)=^𝑁−1_𝑘=01𝑘𝑞𝑘!𝑁−1_𝑘𝐺^(𝑘), (45) where 𝐺^(𝑘)=𝑑^𝑘𝐺^(𝑞)/𝑑𝑞^𝑘 and 𝐺^𝑁(𝑞)=𝐺(𝑞)×⋯×𝐺(𝑞)_𝑁-times. Then 𝐺(𝑞)=1/(𝑞𝑒^𝑡+1) is a solution of (45).

Let us define 𝐺^(𝑘)(𝑡,𝑥)=𝐺^(𝑘)(𝑞)𝑒^𝑥𝑡. Then we obtain the following corollary.

Corollary 2. For 𝑁∈ℕ, one considers 𝐺^𝑁()=1𝑡,𝑥(𝑁−1)!^𝑁−1_𝑘=0()!𝑘𝑁−𝑘−1𝑁−1²𝑞^𝑘𝐺^(𝑘)()=𝑡,𝑥_{𝑁−1𝑘=0}1𝑘𝑞𝑘!𝑁−1_𝑘𝐺^(𝑘)(𝑡,𝑥). (46) Then 𝐺(𝑡,𝑥)=𝑒^𝑥𝑡/(𝑞𝑒^𝑡+1) is a solution of (46).

3. Identities on the High-Order 𝑞-Euler Numbers and Polynomials with Weight 0

From (3), (7), and (8), we get 𝐺^𝑁𝑞=12^𝑁2𝑞𝑒^𝑡2+1×⋯×𝑞𝑒^𝑡+1_𝑁-times=12^𝑁∞∼_𝑛=0𝐸^(𝑁)_𝑛,𝑞𝑡^𝑛,𝐺𝑞=1𝑛!22𝑞𝑒^𝑡=1+12^∞∼_𝑛=0𝐸_𝑛,𝑞𝑡^𝑛.𝑛! (47) From (47), we note that 𝐺^(𝑘)=𝑑^𝑘𝐺𝑞𝑑𝑞^𝑘=12^∞_𝑛=0𝑑𝐸^𝑘∼_𝑛,𝑞𝑑𝑞^𝑘𝑡^𝑛.𝑛! (48) Therefore, by (47), (48), and (45), we obtain the following theorem.

Theorem 3. For 𝑁∈ℕ and 𝑛∈ℕ∪{0}, one has 𝐸^∼(𝑁)_𝑛,𝑞=2^{𝑁−1𝑁−1}_𝑘=01𝑘𝑞𝑘!𝑁−1_𝑘𝑑𝐸^𝑘∼_𝑛,𝑞𝑑𝑞^𝑘. (49)

From (48), we get 𝐺^(𝑘)(𝑡,𝑥)=𝐺^(𝑘)𝑞𝑒^𝑥𝑡=12^∞_𝑛=0𝑑𝐸^𝑘∼_𝑛,𝑞𝑑𝑞^𝑘𝑡^𝑛𝑛!^∞_𝑛=0𝑥^𝑛𝑡^𝑛=𝑛!_{∞𝑛=0𝑛𝑙=0}12𝑛𝑙𝑥^𝑛−𝑙𝑑𝐸^𝑘∼_𝑙,𝑞𝑑𝑞^𝑘𝑡^𝑛.𝑛! (50) Therefore, by (7), (47), and (50), we obtain the following corollary.

Corollary 4. For 𝑁∈ℕ and 𝑛∈ℕ∪{0}, one has 𝐸^∼(𝑁)_𝑛,𝑞(𝑥)=2^{𝑁−1𝑁−1}_𝑘=01𝑘𝑞𝑘!𝑁−1_𝑘^𝑛_𝑙=0𝑛𝑙𝑥^𝑛−𝑙𝑑𝐸^𝑘∼_𝑙,𝑞𝑑𝑞^𝑘. (51)

From (3) and (7), we get ^∞∼_𝑛=𝐸₀^(𝑁)_𝑛,𝑞𝑡^𝑛=2𝑛!𝑞𝑒^𝑡2+1×⋯×𝑞𝑒^𝑡+1_𝑁-times=^∞_𝑙1^∼₌₀𝐸_𝑙1,𝑞𝑡^𝑙1𝑙₁!×⋯×^∞_𝑙𝑁^∼₌₀𝐸_{𝑙𝑁,𝑞}𝑡^𝑙𝑁𝑙_𝑁!=^∞_{𝑛=0𝑙1+⋯+𝑙𝑁=𝑛}𝑛!𝐸^∼_𝑙1,𝑞⋯𝐸^∼_{𝑙𝑁,𝑞}𝑙₁!⋯𝑙_𝑁!𝑡^𝑛=𝑛!_{∞𝑛=0𝑙1+⋯+𝑙𝑁=𝑛}𝑛𝑙₁,…,𝑙_𝑁^∼𝐸_𝑙1,𝑞⋯𝐸^∼_{𝑙𝑁,𝑞}𝑡^𝑛.𝑛! (52) Therefore, by (49) and (52), we obtain the following corollary.

Corollary 5. For 𝑁∈ℕ and 𝑛∈ℕ∪{0}, one has _{𝑙1+⋯+𝑙𝑁=𝑛}𝑛𝑙₁,…,𝑙_𝑁^∼𝐸_𝑙1,𝑞⋯𝐸^∼_{𝑙𝑁,𝑞}=2^{𝑁−1𝑁−1}_𝑘=01𝑘𝑞𝑘!𝑁−1_𝑘𝑑𝐸^𝑘∼_𝑛,𝑞𝑑𝑞^𝑘. (53)

Acknowledgment

The present research has been conducted by the research grant of Kwangwoon University in 2013.