Euler Basis, Identities, and Their Applications

Abstract

Let 𝑉_𝑛={𝑝(𝑥)∈ℚ[𝑥]|deg𝑝(𝑥)≤𝑛} be the (𝑛+1)-dimensional vector space over ℚ. We show that {𝐸₀(𝑥),𝐸₁(𝑥),…,𝐸_𝑛(𝑥)} is a good basis for the space 𝑉_𝑛, for our purpose of arithmetical and combinatorial applications. Thus, if 𝑝(𝑥)∈ℚ[𝑥] is of degree 𝑛, then 𝑝(𝑥)=∑^𝑛𝑙=0𝑏_𝑙𝐸_𝑙(𝑥) for some uniquely determined 𝑏_𝑙∈ℚ. In this paper we develop method for computing 𝑏_𝑙 from the information of 𝑝(𝑥).

1. Introduction

The Euler polynomials, 𝐸_𝑛(𝑥), are given by 2𝑒^𝑡+1𝑒^𝑥𝑡=𝑒^{𝐸(𝑥)𝑡}=^∞_𝑛=0𝐸_𝑛(𝑥)𝑡^𝑛𝑛!,(1.1)(see [1–20]) with the usual convention about replacing 𝐸^𝑛(𝑥) by 𝐸_𝑛(𝑥). In the special case, 𝑥=0,𝐸_𝑛(0)=𝐸_𝑛 are called the 𝑛th Euler numbers. The Bernoulli numbers are also defined by 𝑡𝑒^𝑡−1=𝑒^𝐵𝑡=^∞_𝑛=0𝐵_𝑛𝑡^𝑛𝑛!,(1.2) (see [1–20]) with the usual convention about replacing 𝐵^𝑛 by 𝐵_𝑛. As is well known, the Bernoulli polynomials are given by 𝐵_𝑛(𝑥)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐵_𝑙𝑥^𝑛−𝑙=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐵_𝑛−𝑙𝑥^𝑙,(1.3) (see [9–15]) From (1.1), (1.2), and (1.3), we note that 𝐵_𝑛(1)−𝐵_𝑛=𝛿_1,𝑛,𝐸_𝑛(1)+𝐸_𝑛=2𝛿_0,𝑛,(1.4) where 𝛿_𝑘,𝑛 is the kronecker symbol.

Let 𝑚,𝑛∈ℤ₊ with 𝑚+𝑛≥2. The formula 𝐵_𝑚(𝑥)𝐵_𝑛(𝑥)=_𝑟⎧⎪⎨⎪⎩⎛⎜⎜⎝2𝑚𝑟⎞⎟⎟⎠𝑛+⎛⎜⎜⎝2𝑛𝑟⎞⎟⎟⎠𝑚⎫⎪⎬⎪⎭𝐵_2𝑟𝐵_{𝑚+𝑛−2𝑟}(𝑥)𝑚+𝑛−2𝑟+(−1)^𝑚+1𝑚!𝑛!𝐵_𝑚+𝑛(𝑚+𝑛)!,(1.5) is proved in [4–6]. Let 𝑉_𝑛={𝑝(𝑥)∈ℚ[𝑥]∣deg𝑝(𝑥)≤𝑛} be the (𝑛+1)-dimensional vector space over ℚ. Probably, {1,𝑥,…,𝑥^𝑛} is the most natural basis for this space. But {𝐸₀(𝑥),𝐸₁(𝑥),…,𝐸_𝑛(𝑥)} is also a good basis for the space 𝑉_𝑛, for our purpose of arithmetical and combinatorial applications. Thus, if 𝑝(𝑥)∈ℚ[𝑥] is of degree 𝑛, then 𝑝(𝑥)=^𝑛_𝑙=0𝑏_𝑙𝐸_𝑙(𝑥),(1.6) for some uniquely determined 𝑏_𝑙∈ℚ. Further, 𝑏_𝑘=12𝑘!𝑝^(𝑘)(1)+𝑝^(𝑘)(0)(𝑘=0,1,2,…,𝑛),(1.7) where 𝑝^(𝑘)(𝑥)=𝑑^𝑘𝑝(𝑥)/𝑑𝑥^𝑘. In this paper we develop methods for computing 𝑏_𝑙 from the information of 𝑝(𝑥). Apply these results to arithmetically and combinatorially interesting identities involving 𝐸₀(𝑥),𝐸₁(𝑥),…,𝐸_𝑛(𝑥),𝐵₀(𝑥),…,𝐵_𝑛(𝑥). Finally, we give some applications of those obtained identities.

2. Euler Basis, Identities, and Their Applications

Let us take 𝑝(𝑥) the polynomial of degree 𝑛 as follows: 𝑝(𝑥)=^𝑛_𝑘=0𝐵_𝑘(𝑥)𝐵_𝑛−𝑘(𝑥).(2.1) From (2.1), we have 𝑝^(𝑘)(𝑥)=(𝑛+1)!(𝑛−𝑘+1)!^𝑛_𝑙=𝑘𝐵_𝑙−𝑘(𝑥)𝐵_𝑛−𝑙(𝑥).(2.2) By (1.7) and (2.2), we get 𝑏_𝑘=12𝑘!𝑝^(𝑘)=1(1)+𝑝^(𝑘)(0)2⎛⎜⎜⎝𝑛+𝑘1⎞⎟⎟⎠^𝑛_𝑙=𝑘𝐵_𝑙−𝑘+𝛿_{1,𝑙−𝑘}𝐵_𝑛−𝑙+𝛿_{1,𝑛−𝑙}+𝐵_𝑙−𝑘𝐵_𝑛−𝑙,(2.3) Thus, we have 𝑏_𝑘=⎛⎜⎜⎝𝑛+𝑘1⎞⎟⎟⎠^𝑛_𝑙=𝑘𝐵_𝑙−𝑘𝐵_𝑛−𝑙+𝐵_{𝑛−𝑘−1}𝑏,(0≤𝑘≤𝑛−3),(2.4)_𝑛−2=772𝑛𝑛²−1,𝑏_𝑛=𝑛+1,𝑏_𝑛−1=0.(2.5) By (1.6), (2.1), (2.3), and (2.4), we get ^𝑛_𝑘=0𝐵=_𝑘(𝑥)𝐵_𝑛−𝑘(𝑥)_𝑛−3_𝑘=0⎛⎜⎜⎝𝑛+𝑘1⎞⎟⎟⎠^𝑛_𝑙=𝑘𝐵_𝑙−𝑘𝐵_𝑛−𝑙+𝐵_{𝑛−𝑘−1}𝐸^𝑘(𝑥)+772𝑛𝑛²−1𝐸_𝑛−2(𝑥)+(𝑛+1)𝐸_𝑛(𝑥).(2.6) Let us consider the following triple identities: 𝑝(𝑥)=_{𝑟+𝑠+𝑡=𝑛}𝐵_𝑟(𝑥)𝐵_𝑠(𝑥)𝐵_𝑡(𝑥)=^𝑛_𝑘=0𝑏_𝑘𝐸_𝑘(𝑥),(2.7) where the sum runs over all 𝑟,𝑠,𝑡∈ℤ₊ with 𝑟+𝑠+𝑡=𝑛. Thus, by (2.7), we get 𝑝^(𝑘)(𝑥)=(𝑛+2)(𝑛+1)𝑛(𝑛−1)⋯(𝑛−𝑘+3)_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟(𝑥)𝐵_𝑠(𝑥)𝐵_𝑡(𝑥).(2.8) From (1.7) and (2.8), we have 𝑏_𝑘=12𝑘!𝑝^(𝑘)=(1)+𝑝^(𝑘)(0)_𝑘𝑛+22_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟=_𝑘𝑛+2(1)𝐵_𝑠(1)𝐵_𝑡(1)+𝐵_𝑟𝐵_𝑠𝐵_𝑡22_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝐵_𝑠𝐵_𝑡+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝐵_𝑠𝐵_𝑡+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝛿_1,𝑠𝐵_𝑡+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝐵_𝑠𝛿_1,𝑡+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝛿_1,𝑠𝐵_𝑡+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝐵_𝑠𝛿_1,𝑡+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝛿_1,𝑠𝛿_1,𝑡+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝛿_1,𝑠𝛿_1,𝑡.(2.9) Therefore, by (2.7) and (2.9), we obtain the following theorem.

Theorem 2.1. For 𝑟,𝑠,𝑡∈ℤ₊, and 𝑛∈ℕ with 𝑛≥3, one has _𝑟+_{𝑠+𝑡=𝑛}=𝐵1_𝑟(𝑥)𝐵_𝑠(𝑥)𝐵_𝑡(𝑥)2^𝑛−2_𝑘=0⎛⎜⎜⎝𝑛+𝑘2⎞⎟⎟⎠2_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝐵_𝑠𝐵_𝑡+3_{𝑟+𝑠=𝑛−𝑘−1}𝐵_𝑟𝐵_𝑠+3𝐵_{𝑛−𝑘−2}+𝛿_{𝑘,𝑛−3}𝐸^𝑘+⎛⎜⎜⎝𝑛+22⎞⎟⎟⎠𝐸_𝑛(𝑥)(𝑥).(2.10)

Let us take the polynomial 𝑝(𝑥) as follows: 𝑝(𝑥)=_{𝑟+𝑠+𝑡=𝑛}𝐵_𝑟(𝑥)𝐵_𝑠(𝑥)𝐸_𝑡(𝑥).(2.11) Then, by (2.11), we get 𝑝^(𝑘)(𝑥)=(𝑛+2)(𝑛+1)𝑛(𝑛−1)⋯(𝑛−𝑘+3)_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟(𝑥)𝐵_𝑠(𝑥)𝐸_𝑡(𝑥).(2.12)

From (1.6), (1.7), and (2.12), we have 𝑏_𝑘=12𝑘!𝑝^(𝑘)(1)+𝑝^(𝑘)(0)=^𝑘𝑛+22_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟=_𝑘𝑛+2(1)𝐵_𝑠(1)𝐸_𝑡(1)+𝐵_𝑟𝐵_𝑠𝐸_𝑡2_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟=_𝑘𝑛+2+𝛿_1,𝑟𝐵_𝑠+𝛿_1,𝑠−𝐸_𝑡+2𝛿_0,𝑡+𝐵_𝑟𝐵_𝑠𝐸_𝑡2−_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝐵_𝑠𝐸_𝑡−_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝛿_1,𝑠𝐸_𝑡+2_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝐵_𝑠𝛿_0,𝑡−_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝛿_1,𝑠𝐸_𝑡+2_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝐵_𝑠𝛿_0,𝑡+2_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝛿_1,𝑠𝛿_0,𝑡+2_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝛿_1,𝑠𝛿_0,𝑡.(2.13) Note that 𝑏_𝑛−1==1⎛⎜⎜⎝𝑛𝑛+−21⎞⎟⎟⎠−_𝑠+𝑡=0𝐵_𝑠𝐸_𝑡−_𝑟+𝑡=0𝐵_𝑟𝐸_𝑡+2_𝑟+𝑠=1𝐵_𝑟𝐵_𝑠−0+2𝐵₀+2𝐵₀+2⋅02⎛⎜⎜⎝𝑛𝑛+−21⎞⎟⎟⎠−1−1+2𝐵¹+𝐵₁+2+2=0.(2.14) Therefore, we obtain the following theorem.

Theorem 2.2. For 𝑛∈ℕ with 𝑛≥2, one has _𝑟+_{𝑠+𝑡=𝑛}=𝐵1_𝑟(𝑥)𝐵_𝑠(𝑥)𝐸_𝑡(𝑥)2^𝑛−2_𝑘=0⎛⎜⎜⎝𝑛+𝑘2⎞⎟⎟⎠2_{𝑟+𝑠=𝑛−𝑘}𝐵_𝑟𝐵_𝑠−2_{𝑟+𝑡=𝑛−𝑘−1}𝐵_𝑟𝐸_𝑡−𝐸_{𝑛−𝑘−2}+4𝐵_{𝑛−𝑘−1}+2𝛿_{𝑘,𝑛−2}𝐸^𝑘+⎛⎜⎜⎝𝑛+22⎞⎟⎟⎠𝐸_𝑛(𝑥)(𝑥).(2.15)

Remark 2.3. By the same method, we obtain the following identities.
(I) _𝑟+_{𝑠+𝑡=𝑛}=𝐵1_𝑟(𝑥)𝐸_𝑠(𝑥)𝐸_𝑡(𝑥)2^𝑛−2_𝑘=0⎛⎜⎜⎝𝑛+𝑘2⎞⎟⎟⎠2_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝐸_𝑠𝐸_𝑡+_{𝑠+𝑡=𝑛−𝑘−1}𝐸_𝑠𝐸_𝑡−4_{𝑟+𝑠=𝑛−𝑘}𝐵_𝑟𝐸_𝑠+4𝐵_𝑛−𝑘−4𝐸_{𝑛−𝑘−1}𝐸^𝑘+⎛⎜⎜⎝𝑛+22⎞⎟⎟⎠𝐸_𝑛(𝑥)(𝑥).(2.16)
(II) _𝑟+_{𝑠+𝑡=𝑛}=𝐸3_𝑟(𝑥_𝑛−2)𝐸_𝑠(𝑥)𝐸_𝑡(𝑥)_𝑘=0⎛⎜⎜⎝𝑛+𝑘2⎞⎟⎟⎠_{𝑟+𝑠=n−𝑘}𝐸_𝑟𝐸_𝑠−2𝐸_𝑛−𝑘𝐸^𝑘(𝑥)+⎛⎜⎜⎝𝑛+22⎞⎟⎟⎠𝐸_𝑛(𝑥).(2.17) Let us consider the polynomial 𝑝(𝑥) as follows: 𝑝(𝑥)=_{𝑟+𝑠+𝑡=𝑛}𝐵_𝑟(𝑥)𝐵_𝑠(𝑥)𝑥^𝑡.(2.18) Thus, by (2.18), we get 𝑝^(𝑘)(𝑥)=(𝑛+2)(𝑛+1)𝑛(𝑛−1)⋯(𝑛−𝑘+3)_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟(𝑥)𝐵_𝑠(𝑥)𝑥^𝑡.(2.19) From (1.6), (1.7), (2.18), and (2.19), we have 𝑏_𝑘=12𝑘!𝑝^(𝑘)(1)+𝑝^(𝑘)(0)=^𝑘𝑛+22_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟=_𝑘𝑛+2(1)𝐵_𝑠(1)+𝐵_𝑟𝐵_𝑠0^𝑡2_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟=_𝑘𝑛+2+𝛿_1,𝑟𝐵_𝑠+𝛿_1,𝑠+𝐵_𝑟𝐵_𝑠0^𝑡2_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝐵_𝑠+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝛿_1,𝑠+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝐵_𝑠+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝛿_1,𝑟𝛿_1,𝑠+_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝐵_𝑠0^𝑡(2..20) Here we note that _𝑟+𝑠+_{𝑡=𝑛−𝑘}𝐵_𝑟𝐵_𝑠=^𝑛−𝑘_𝑡=0_{𝑟+𝑠=𝑛−𝑘−𝑡}𝐵_𝑟𝐵_𝑠=^𝑛−𝑘_𝑡=0_{𝑟+𝑠=𝑡}𝐵_𝑟𝐵_{𝑠𝑟+𝑠+}_{𝑡=𝑛−𝑘}𝐵_𝑟𝛿_1,𝑠=⎧⎪⎨⎪⎩^{𝑛−𝑘−1}0,∑_𝑟=0𝐵_𝑟,iiff𝑘𝑘≤=𝑛𝑛,_{𝑟+𝑠+𝑡=𝑛−𝑘}−1,𝐵_𝑠𝛿_1,𝑟=⎧⎪⎨⎪⎩^{𝑛−𝑘−1}0,∑_𝑟=0𝐵_𝑟,iiff𝑘𝑘≤=𝑛𝑛,_{𝑟+𝑠+𝑡=𝑛−𝑘}−1,𝛿_1,𝑟𝛿_1,𝑠=10,,iiff𝑘𝑘≤=𝑛𝑛−−21,or𝑛,_{𝑟+𝑠+𝑡=𝑛−𝑘}𝐵_𝑟𝐵_𝑠0^𝑡=_{𝑟+𝑠=𝑛−𝑘}𝐵_𝑟𝐵_𝑠,∀𝑘.(2.21) It is easy to show that 𝑏_𝑛−1=12⎛⎜⎜⎝𝑛𝑛+−21⎞⎟⎟⎠_𝑟+𝑠=0𝐵_𝑟𝐵_𝑠+2_𝑟+𝑠=1𝐵_𝑟𝐵_𝑠+2𝐵₀=12⎛⎜⎜⎝𝑛𝑛+−21⎞⎟⎟⎠1+2𝐵¹+𝐵₂+2=12⎛⎜⎜⎝𝑛𝑛+−21⎞⎟⎟⎠.(2.22) Therefore, by (1.6), (2.18), (2.20), and (2.22), we obtain the following theorem.

Theorem 2.4. For 𝑛∈ℕ with 𝑛≥2, one has _𝑟+_{𝑠+𝑡=𝑛}𝐵_𝑟(𝑥)𝐵_𝑠(𝑥)𝑥^𝑡=12^𝑛−2_𝑘=0⎛⎜⎜⎝𝑛+𝑘2⎞⎟⎟⎠^{𝑛−𝑘−1}_𝑡=0_{𝑟+𝑠=𝑡}𝐵_𝑟𝐵_𝑠+2_{𝑟+𝑠=𝑛−𝑘}𝐵_𝑟𝐵_𝑠+2^{𝑛−𝑘−1}_𝑟=0+1𝐵_𝑟+1𝐸_𝑘(𝑥)2⎛⎜⎜⎝𝑛𝑛+−21⎞⎟⎟⎠𝐸^𝑛−1(𝑥)+⎛⎜⎜⎝𝑛+𝑛2⎞⎟⎟⎠𝐸_𝑛(𝑥).(2.23)

Remark 2.5. By the same method, we can derive the following identities.
(I) _𝑟+_{𝑠+𝑡=𝑛}𝐵_𝑟(𝑥)𝐸_𝑠(𝑥)𝑥^𝑡=12^𝑛−2_𝑘=0⎛⎜⎜⎝𝑛+𝑘2⎞⎟⎟⎠−^{𝑛−𝑘−1}_𝑡=0_{𝑟+𝑠=𝑡}𝐵_𝑟𝐸_𝑠−^{𝑛−𝑘−1}_𝑠=0𝐸_𝑠+2^𝑛−𝑘_𝑟=0+1𝐵_𝑟+2𝐸_𝑘(𝑥)2⎛⎜⎜⎝𝑛𝑛+−21⎞⎟⎟⎠𝐸^𝑛−1(𝑥)+⎛⎜⎜⎝𝑛+𝑛2⎞⎟⎟⎠𝐸_𝑛(𝑥).(2.24)
(II) _𝑟+_{𝑠+𝑡=𝑛}𝐸_𝑟(𝑥)𝐸_𝑠(𝑥)𝑥^𝑡=12^𝑛−2_𝑘=0⎛⎜⎜⎝𝑛+𝑘2⎞⎟⎟⎠^{𝑛−𝑘−1}_𝑡=0_{𝑟+𝑠=𝑡}𝐸_𝑟𝐸_𝑠+2_{𝑟+𝑠=𝑛−𝑘}𝐸_𝑟𝐸_𝑠−4^𝑛−𝑘_𝑟=0+1𝐸_𝑟+4𝐸_𝑘(𝑥)2⎛⎜⎜⎝𝑛𝑛+−21⎞⎟⎟⎠𝐸^𝑛−1(𝑥)+⎛⎜⎜⎝𝑛+22⎞⎟⎟⎠𝐸_𝑛(𝑥).(2.25)

Now we generalize the above consideration to the completely arbitrary case. Let 𝑝(𝑥)=_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥),(2.26) where the sum runs over all nonnegative integers 𝑖₁,𝑖₂,…,𝑖_𝑟,𝑗₁,…,𝑗_𝑠 satisfying 𝑖₁+𝑖₂+⋯+𝑖_𝑟+𝑗₁+⋯+𝑗_𝑠=𝑛. From (2.26), we note that 𝑝^(𝑘)(𝑥)=(𝑛+𝑟+𝑠−1)⋯(𝑛+𝑟+𝑠−𝑘)_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)×𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥).(2.27)

By (1.6), (1.7), (2.18), and (2.27), we get 𝑏_𝑘=12𝑘!𝑝^(𝑘)=1(1)+𝑝^(𝑘)(0)2⎛⎜⎜⎝𝑛+𝑟+𝑘𝑠−1⎞⎟⎟⎠_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}𝐵_𝑖1=1(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)+𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠2×⎛⎜⎜⎝𝑛+𝑟+𝑘𝑠−1⎞⎟⎟⎠_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}×𝐵−_𝑖1𝐸+_𝑗1𝛿_1,𝑖1⋯𝐵_𝑖𝑟+𝛿_1,𝑖𝑟=1+2𝛿_0,𝑗1⋯−𝐸_j𝑠+2𝛿_0,𝑗𝑠+𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠2⎛⎜⎜⎝𝑛+𝑟+𝑘𝑠−1⎞⎟⎟⎠⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑎≥𝑘+𝑟−𝑛}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑛+𝑎−𝑘−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭.(2.28) Note that 𝑏_𝑛=12⎛⎜⎜⎝𝑛+𝑟+𝑛𝑠−1⎞⎟⎟⎠⎧⎪⎨⎪⎩_{0≤𝑐≤𝑠}⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑐=0}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=0}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠=1⎫⎪⎬⎪⎭2⎛⎜⎜⎝𝑛+𝑟+𝑛𝑠−1⎞⎟⎟⎠((2−1)^𝑠𝑏+1)=⎛⎜⎜⎝𝑛+𝑟+𝑛𝑠−1_𝑛−1⎞⎟⎟⎠,=12⎛⎜⎜⎝𝑛+𝑛𝑟+−𝑠1−1⎞⎟⎟⎠⎧⎪⎨⎪⎩_{𝑟−1≤𝑎≤𝑟0≤𝑐≤𝑠}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=1+𝑎−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=1}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠=1⎫⎪⎬⎪⎭2⎛⎜⎜⎝𝑛+𝑛𝑟+−𝑠1−1⎞⎟⎟⎠⎧⎪⎨⎪⎩𝑟(2−1)^𝑠+_{0≤𝑐≤𝑠}⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐−1=12(𝑟+𝑐)−21(𝑟+𝑠)⎫⎪⎬⎪⎭2⎛⎜⎜⎝𝑛+𝑛𝑟+−𝑠1−1⎞⎟⎟⎠𝑟−12𝑟+21𝑠−12𝑟−21𝑠=0.(2.29) Therefore, by (1.6), (2.28), and (2.29), we obtain the following theorem.

Theorem 2.6. For 𝑛∈ℕ with 𝑛≥2, one has _{𝑖1+⋯+𝑖𝑟+}_{𝑗1+⋯+𝑗𝑠=𝑛}=1𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)2^𝑛−2_𝑘=0⎛⎜⎜⎝𝑛+𝑟+𝑘𝑠−1⎞⎟⎟⎠⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑎≥𝑘+𝑟−𝑛}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑛+𝑎−𝑘−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭𝐸_𝑘+⎛⎜⎜⎝𝑛+𝑟+𝑛𝑠−1⎞⎟⎟⎠𝐸_𝑛(𝑥)(𝑥).(2.30)

Let us consider the polynomial 𝑝(𝑥) of degree 𝑛 as 𝑝(𝑥)=_{𝑡+𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑥^𝑡.(2.31) Then, from (2.31), we have 𝑝^(𝑘)(𝑥)=(×𝑛+𝑟+𝑠)(𝑛+𝑟+𝑠−1)⋯(𝑛+𝑟+𝑠−𝑘+1)_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛−𝑘}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑥^𝑡.(2.32) By (1.7) and (2.32), we get 𝑏_𝑘=12𝑘!𝑝^(𝑘)=1(1)+𝑝^(𝑘)(0)2⎛⎜⎜⎝𝑛+𝑘𝑟+𝑠⎞⎟⎟⎠_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛−𝑘}𝐵_𝑖1=1(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)+𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠0^𝑡2×⎛⎜⎜⎝𝑛+𝑘𝑟+𝑠⎞⎟⎟⎠_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛−𝑘}×𝐵−_𝑖1𝐸+_𝑗0𝛿_1,𝑖1⋯𝐵_𝑖𝑟+𝛿_1,𝑖𝑟+2𝛿_0,𝑗1⋯−𝐸_𝑗𝑠+2𝛿_1,𝑗𝑠+𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠0^𝑡(2.33) From (2.33), we can derive the following equation: 𝑏_𝑘=12⎛⎜⎜⎝𝑛+𝑘𝑟+𝑠⎞⎟⎟⎠⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑎≥𝑘+𝑟−𝑛}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×^{𝑛+𝑎−𝑘−𝑟}_𝑡=0_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑡}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭.(2.34) Observe now that 𝑏_𝑛=12⎛⎜⎜⎝𝑛+𝑛𝑟+𝑠⎞⎟⎟⎠⎧⎪⎨⎪⎩^𝑠_𝑐=0⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑐=0}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=0}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠=1⎫⎪⎬⎪⎭2⎛⎜⎜⎝𝑛+𝑛𝑟+𝑠⎞⎟⎟⎠(2−1)^𝑠𝑏+1=⎛⎜⎜⎝𝑛+𝑛𝑟+𝑠⎞⎟⎟⎠,(2.35)_𝑛−1=12⎛⎜⎜⎝𝑛+𝑛−𝑟+1𝑠⎞⎟⎟⎠⎧⎪⎨⎪⎩_{𝑟−1≤𝑎≤𝑟0≤𝑐≤𝑠}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×^{1+𝑎−𝑟}_𝑡=0_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑡}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=1}𝐵_𝑖1⋯𝐵_i𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠=1⎫⎪⎬⎪⎭2⎛⎜⎜⎝𝑛+𝑛−𝑟+1𝑠⎞⎟⎟⎠𝑟+1−12𝑟+21𝑠−12𝑟−21𝑠=12⎛⎜⎜⎝𝑛+𝑛−𝑟+1𝑠⎞⎟⎟⎠.(2.36) Therefore, by (1.6), (2.31), (2.34), (2.35), and (2.36), we obtain the following theorem.

Theorem 2.7. For 𝑛∈ℕ with 𝑛≥2, one has _{𝑖1+⋯+𝑖𝑟+}_{𝑗1+⋯+𝑗𝑠+𝑡=𝑛}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑥^𝑡=12^𝑛−2_𝑘=0⎛⎜⎜⎝𝑛+𝑘𝑟+𝑠⎞⎟⎟⎠⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑎≥𝑘+𝑟−𝑛}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×^{𝑛+𝑎−𝑘−𝑟}_𝑡=0_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=t}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭𝐸_𝑘+1(𝑥)2⎛⎜⎜⎝𝑛+𝑛−𝑟+1𝑠⎞⎟⎟⎠𝐸^𝑛−1(𝑥)+⎛⎜⎜⎝𝑛+𝑛𝑟+𝑠⎞⎟⎟⎠𝐸_𝑛(𝑥).(2.37)

Let us consider the following polynomial of degree 𝑛. 𝑝(𝑥)=_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛}1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝐵^𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥).(2.38) Thus, by (2.38), we get 𝑝^(𝑘)(𝑥)=(𝑟+𝑠)^𝑘_{𝑖1+⋯+𝑖𝑟+𝑗}_{1+⋯+𝑗𝑠=𝑛−𝑘}1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!×𝐵^𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥).(2.39) From (1.7), we have 𝑏_𝑘=12𝑘!𝑝^(𝑘)=(𝑟+𝑠)_𝑘(1)+𝑝^(𝑘)(0)2𝑘!_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}1×𝐵_𝑖1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!=(𝑟+𝑠)_𝑘(1)⋯𝐵_𝑖𝑟(1)×𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)+𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠2𝑘!_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}1×𝐵_𝑖1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!+𝛿_1,𝑖1⋯𝐵_𝑖𝑟+𝛿_1,𝑖𝑟×−𝐸_𝑗1+2𝛿_0,𝑗1⋯−𝐸_𝑗𝑠+2𝛿_0,𝑗𝑠+𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠.(2.40) Thus, by (2.40), we get 𝑏_𝑘=(𝑟+𝑠)^𝑘2𝑘!⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑎≥𝑘+𝑟−𝑛}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^s−𝑐×_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑛+𝑎−𝑘−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}𝑖₁!⋯𝑖_𝑎!𝑗₁!⋯𝑗_𝑐!𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!⎫⎪⎪⎬⎪⎪⎭.(2.41) Now, we note that 𝑏_𝑛=(𝑟+𝑠)^𝑛2𝑛!⎧⎪⎨⎪⎩^𝑠_𝑐=0⎛⎜⎜⎝𝑐s⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑐=0}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑐𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑐!+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=0}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠=(𝑟+𝑠)_𝑛𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!⎫⎪⎬⎪⎭2𝑛!(2−1)^𝑠+1=(𝑟+𝑠)^𝑛𝑏𝑛!_𝑛−1,=(𝑟+𝑠)^𝑛−12(𝑛−1)!⎧⎪⎨⎪⎩_{𝑟−1≤𝑎≤𝑟0≤𝑐≤𝑠}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=1+𝑎−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐𝑖₁!⋯𝑖_𝑎!𝑗₁!⋯𝑗_𝑐!+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=1}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠=(𝑟+𝑠)_𝑛−1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!⎫⎪⎬⎪⎭2(𝑛−1)!⎧⎪⎨⎪⎩𝑟(2−1)^𝑠+^𝑠_𝑐=0⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐−12(𝑟+𝑐)−21(𝑟+𝑠)⎫⎪⎬⎪⎭=0.(2.42) Therefore, by (1.6), (2.38), (2.41), and (2.42), we obtain the following theorem.

Theorem 2.8. For 𝑛∈ℕ with 𝑛≥2, one has _{𝑖1+⋯+𝑖𝑟+}_{𝑗1+⋯+𝑗𝑠=𝑛}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)=1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!2^𝑛−2_𝑘=0(𝑟+𝑠)^𝑘𝑘!⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑎≥𝑘+𝑟−𝑛}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑛+𝑎−𝑘−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}𝑖₁!⋯𝑖_𝑎!𝑗₁!⋯𝑗_𝑐!𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!⎫⎪⎪⎬⎪⎪⎭𝐸^𝑘+(𝑟+𝑠)_𝑛(𝑥)𝑛!𝐸^𝑛(𝑥).(2.43)

By the same method, we can obtain the following identity: _{𝑖1+⋯+𝑖𝑟+}_{𝑗1+⋯+𝑗𝑠+𝑡=𝑛}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑥^𝑡=1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!2^𝑛−2_𝑘=0(𝑟+𝑠+1)^𝑘𝑘!⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑎≥𝑘+𝑟−𝑛}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×^{𝑛+𝑎−𝑘−𝑟}_𝑡=01(𝑛+𝑎−𝑘−𝑟−𝑡)!_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑡}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐+_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘}𝑖₁!⋯𝑖_𝑎!𝑗₁!⋯𝑗_𝑐!𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!⎫⎪⎪⎬⎪⎪⎭𝐸^𝑘+(𝑟+𝑠+1)_𝑛−1(𝑥)2(𝑛−1)!𝐸^𝑛−1(𝑥)+(𝑟+𝑠+1)^𝑛𝑛!𝐸^𝑛(𝑥).(2.44)

Acknowledgments

This paper 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.