Bernoulli Basis and the Product of Several Bernoulli Polynomials

Abstract

We develop methods for computing the product of several Bernoulli and Euler polynomials by using Bernoulli basis for the vector space of polynomials of degree less than or equal to 𝑛.

1. Introduction

It is well known that, the 𝑛th Bernoulli and Euler numbers are defined by ^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐵_𝑙−𝐵_𝑛=𝛿_1,𝑛,^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐸_𝑙+𝐸_𝑛=2𝛿_0,𝑛,(1.1) where 𝐵₀=𝐸₀=1 and 𝛿_𝑘,𝑛 is the Kronecker symbol (see [1–20]).

The Bernoulli and Euler polynomials are also defined by 𝐵_𝑛(𝑥)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐵_𝑛−𝑙𝑥^𝑙,𝐸_𝑛(𝑥)=^𝑛_𝑙=0⎛⎜⎜⎝𝑛𝑙⎞⎟⎟⎠𝐸_𝑛−𝑙𝑥^𝑙.(1.2) Note that {𝐵₀(𝑥),𝐵₁(1),…,𝐵_𝑛(𝑥)} forms a basis for the space ℙ_𝑛={𝑝(𝑥)∈ℚ[𝑥]∣deg𝑝(𝑥)≤𝑛}.

So, for a given 𝑝(𝑥)∈ℙ_𝑛, we can write 𝑝(𝑥)=^𝑛_𝑘=0𝑎_𝑘𝐵_𝑘(𝑥),(1.3) (see [8–12]) for uniquely determined 𝑎_𝑘∈ℚ.

Further, 𝑎_𝑘=1𝑘!𝑝^(𝑘−1)(1)−𝑝^(𝑘−1)(0),where𝑝^(𝑘)(𝑥)=𝑑^𝑘𝑝(𝑥)𝑑𝑥^𝑘𝑎₀,=¹⁰𝑝(𝑡)𝑑𝑡,where𝑘=1,2,…,𝑛.(1.4) Probably, {1,𝑥,…,𝑥^𝑛} is the most natural basis for the space ℙ_𝑛. But {𝐵₀(𝑥),𝐵₁(𝑥),…,𝐵_𝑛(𝑥)} is also a good basis for the space ℙ_𝑛, for our purpose of arithmetical and combinatorial applications.

What are common to 𝐵_𝑛(𝑥), 𝐸_𝑛(𝑥), 𝑥^𝑛? A few proportion common to them are as follows: (i)they are all monic polynomials of degree 𝑛 with rational coefficients; (ii)(𝐵_𝑛(𝑥))′=𝑛𝐵_𝑛−1(𝑥), (𝐸_𝑛(𝑥))′=𝑛𝐸_𝑛−1(𝑥), (𝑥^𝑛)′=𝑛𝑥^𝑛−1; (iii)∫𝐵_𝑛(𝑥)𝑑𝑥=𝐵_𝑛+1(𝑥)/(𝑛+1)+𝑐, ∫𝐸_𝑛(𝑥)𝑑𝑥=𝐸_𝑛+1(𝑥)/(𝑛+1)+𝑐, ∫𝑥^𝑛𝑑𝑥=𝑥^𝑛+1/(𝑛+1)+𝑐.

In [5, 6], Carlitz introduced the identities of the product of several Bernoulli polynomials: 𝐵_𝑚(𝑥)𝐵_𝑛(𝑥)=^∞_𝑟=0⎧⎪⎨⎪⎩⎛⎜⎜⎝2𝑚𝑟⎞⎟⎟⎠𝑛+⎛⎜⎜⎝2𝑛𝑟⎞⎟⎟⎠𝑚⎫⎪⎬⎪⎭𝐵_2𝑟𝐵_{𝑚+𝑛−2𝑟}(𝑥)𝑚+𝑛−2𝑟+(−1)^𝑚+1×𝑚!𝑛!(𝑚+𝑛)!𝐵^𝑚+𝑛(𝑚+𝑛≥2),₁₀𝐵_𝑚(𝑥)𝐵_𝑛(𝑥)𝐵_𝑝(𝑥)𝐵_𝑞(𝑥)𝑑𝑥=(−1)^{∞𝑚+𝑛+𝑝+𝑞}_𝑟_,𝑠=0×(𝑚+𝑛−2𝑟−⎧⎪⎨⎪⎩1⎛⎜⎜⎝)2𝑚!𝑟(⎞⎟⎟⎠𝑝𝑛++𝑞⎛⎜⎜⎝−2𝑛𝑟2⎞⎟⎟⎠𝑠𝑚−⎫⎪⎬⎪⎭1⎧⎪⎨⎪⎩)⎛⎜⎜⎝!2𝑝𝑠⎞⎟⎟⎠𝑞+⎛⎜⎜⎝2𝑞𝑠⎞⎟⎟⎠𝑝⎫⎪⎬⎪⎭(𝑚+𝑛+𝑝+𝑞−2𝑟−2𝑠)!𝐵^𝑟𝐵_𝑠𝐵_{𝑚+𝑛+𝑝+𝑞−2𝑟−2}+(−1)^𝑚+𝑝𝑚!𝑛!(𝑚+𝑛)!(𝑝𝑝+!𝑞𝑞!)!𝐵^𝑚+𝑛𝐵_𝑝+𝑞.(1.5) In this paper, we will use (1.4) to derive the identities of the product of several Bernoulli and Euler polynomials.

2. The Product of Several Bernoulli and Euler Polynomials

Let us consider the following polynomials of degree 𝑛: 𝑝(𝑥)=_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥),(2.1) where the sum runs over all nonnegative integers 𝑖₁,…,𝑖_𝑟,  𝑗₁,…𝑗_𝑠 satisfying 𝑖₁+⋯+𝑖_𝑟+𝑗₁+⋯+𝑗_𝑠=𝑛,  𝑟+𝑠=1, 𝑟,𝑠≥0.

Thus, from (2.1), we have 𝑝^(𝑘)(𝑥)=(×𝑛+𝑟+𝑠−1)(𝑛+𝑟+𝑠−2)⋯(𝑛+𝑟+𝑠−𝑘)_{∞𝑖1+⋯+𝑖𝑟+𝑗}_{1+⋯+𝑗𝑠=𝑛−𝑘}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥).(2.2) For 𝑘=1,2,…,𝑛, by (1.4), we get 𝑎_𝑘=1𝑘!𝑝^(𝑘−1)=(1)−𝑝^(𝑘−1)(0)_{𝑘𝑛+𝑟+𝑠}𝑛+𝑟+𝑠_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘+1}𝐵_𝑖1=_{𝑘𝑛+𝑟+𝑠}(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠𝑛+𝑟+𝑠⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑘+𝑟−𝑛−1≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×^∞_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯}_{+𝑗𝑠=𝑛+𝑎+1−𝑘−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐−_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘+1}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭.(2.3) From (2.3), we note that 𝑎_𝑛=(^{𝑛𝑛+𝑟+𝑠})𝑛+𝑟+𝑠_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=1}𝐵_𝑖1=(_{𝑛𝑛+𝑟+𝑠}(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠)=𝑛(+𝑟+𝑠−12+1𝑟−−12𝑠−−21(𝑟+𝑠)_{𝑛𝑛+𝑟+𝑠})𝑎𝑛+𝑟+𝑠(𝑟+𝑠)=⎛⎜⎜⎝𝑛+𝑟+𝑛𝑠−1_𝑛−1⎞⎟⎟⎠,=1𝑛×+𝑟+𝑠⎛⎜⎜⎝𝑛+𝑛−𝑟+1_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=2}𝑠⎞⎟⎟⎠=1𝐵_𝑖1(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠𝑎𝑛+𝑟+𝑠⎛⎜⎜⎝𝑛+𝑛−𝑟+1𝑠⎞⎟⎟⎠⎧⎪⎨⎪⎩16𝑟+2121⎛⎜⎜⎝𝑟+2𝑠⎞⎟⎟⎠−61𝑟−−21−12⎛⎜⎜⎝𝑟+2𝑠⎞⎟⎟⎠⎫⎪⎬⎪⎭=0,₀==¹⁰𝑝(𝑡)𝑑𝑡_∞_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠𝑖=𝑛}¹_𝑙1=0⋯^𝑖𝑟_{𝑙𝑟𝑗=0}¹_𝑝1=0⋯^𝑗𝑠_𝑝_𝑠=0⎛⎜⎜⎝𝑖₁𝑙₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑖_𝑟𝑙_𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑗₁𝑝₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑗_𝑠𝑝_𝑠×𝐵_{𝑖1−𝑙1}⎞⎟⎟⎠⋯𝐵_{𝑖𝑟−𝑙𝑟}𝐸_{𝑗1−𝑝1}𝐸_{𝑗𝑠−𝑝𝑠}𝑙₁+⋯+𝑙_𝑟+𝑝₁+⋯+𝑝_𝑠+1.(2.4) Therefore, by (1.3), (2.1), (2.3), and (2.4), we obtain the following theorem.

Theorem 2.1. For 𝑛∈ℕ with 𝑛≥2, we have _{𝑖1+⋯+𝑖𝑟+}_{𝑗1+⋯+𝑗𝑠=𝑛}=1𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑛+𝑟+𝑠^𝑛−2_𝑘=1×⎧⎪⎪⎨⎪⎪⎩⎛⎜⎜⎝𝑛+𝑘𝑟+𝑠_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑘+𝑟−𝑛−1≤𝑎≤𝑟}⎞⎟⎟⎠⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯}_{+𝑗𝑐=𝑛+𝑎+1−𝑘−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐−^∞_{𝑖1+⋯+𝑖𝑟+𝑗1}_{+⋯+𝑗𝑠=𝑛−𝑘+1}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑘1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭𝐵_𝑘+(𝑥)+⎛⎜⎜⎝𝑛+𝑟+𝑛𝑠−1⎞⎟⎟⎠𝐵_𝑛(𝑥)_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠𝑖=𝑛1}_𝑙1=0…^𝑖𝑟_{𝑙𝑟𝑗=0}¹_𝑝1=0⋯^𝑗𝑠_𝑝_𝑠=0⎛⎜⎜⎝𝑖₁𝑙₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑖_𝑟𝑙_𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑗₁𝑝₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑗_𝑠𝑝_𝑠×𝐵_{𝑖1−𝑙1}⎞⎟⎟⎠⋯𝐵_{𝑖𝑟−𝑙𝑟}𝐸_{𝑗1−𝑝1}𝐸_{𝑗𝑠−𝑝𝑠}𝑙₁+⋯+𝑙_𝑟+𝑝₁+⋯𝑝_𝑠+1.(2.5)

Let us take the polynomial 𝑝(𝑥) of degree 𝑛 as follows: 𝑝(𝑥)=_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑥^𝑡,(2.6) Then, from (2.6), 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,(2.7) By (1.4) and (2.7), we get, for 𝑘=1,2,…,𝑛, 𝑎_𝑘=1𝑘!𝑝^(𝑘−1)=1(1)−𝑝^(𝑘−1)(0)𝑛×+𝑟+𝑠+1⎛⎜⎜⎝𝑛+𝑟+𝑘𝑠+1_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛−𝑘+1}⎞⎟⎟⎠=_{𝑘𝑛+𝑟+𝑠+1}𝐵_𝑖1(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠0^𝑡𝑛+𝑟+𝑠+1⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑘+𝑟−𝑛−1≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×^{𝑛+𝑎+1−𝑘−𝑟}_𝑡=0_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑡}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐−_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘+1}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭,(2.8) Now, we look at 𝑎_𝑛 and 𝑎_𝑛−1. 𝑎_𝑛=^{𝑛𝑛+𝑟+𝑠+1}𝑛+𝑟+𝑠+1_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=1}𝐵_𝑖1=_{𝑛𝑛+𝑟+𝑠+1}(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠0^𝑡=𝑛+𝑟+𝑟+𝑠+𝑠+1112(𝑟+𝑠)+1−−12(𝑟+𝑠)𝑎𝑛+𝑟+𝑠+1⎛⎜⎜⎝𝑛+𝑟+𝑛𝑠+1⎞⎟⎟⎠=⎛⎜⎜⎝𝑛+𝑛𝑟+𝑠_𝑛−1⎞⎟⎟⎠,=^{𝑛+𝑟+𝑠+1𝑛−1}𝑛+𝑟+𝑠+1_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=2}𝐵_𝑖1=_{𝑛+𝑟+𝑠+1𝑛−1}(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠0^𝑡=𝑛+𝑟+1𝑠+1⎧⎪⎨⎪⎩61𝑟+1+2121⎛⎜⎜⎝𝑟+2𝑠⎞⎟⎟⎠+21(𝑟+𝑠)−61𝑟−−21−12⎛⎜⎜⎝𝑟+2𝑠⎞⎟⎟⎠⎫⎪⎬⎪⎭𝑛+𝑟+𝑠+1⎛⎜⎜⎝𝑛+𝑛𝑟+−𝑠1+1⎞⎟⎟⎠𝑟+𝑠2+2=21⎛⎜⎜⎝𝑛+𝑛−𝑟+1𝑠⎞⎟⎟⎠,(2.9) From (2.6), we note that 𝑎₀==¹⁰𝑝(𝑡)𝑑𝑡_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠𝑖+𝑡=𝑛1}_𝑙1=0⋯^𝑖𝑟_{𝑙𝑟𝑗=0}¹_𝑝1=0⋯^𝑗𝑠_𝑝_𝑠=0⎛⎜⎜⎝𝑖₁𝑙₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑖_𝑟𝑙_𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑗₁𝑝₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑗_𝑠𝑝_𝑠×𝐵_{𝑖1−𝑙1}⎞⎟⎟⎠⋯𝐵_{𝑖𝑟−𝑙𝑟}𝐸_{𝑗1−𝑝1}𝐸_{𝑗𝑠−𝑝𝑠}1𝑙₁+⋯+𝑙_𝑟+𝑝₁+⋯𝑝_𝑠+𝑡+1.(2.10) Therefore, by (1.3), (2.6), (2.8), (2.9), and (2.10), we obtain the following theorem.

Theorem 2.2. 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=1𝑛+𝑟+𝑠+1^𝑛−2_𝑘=1×⎧⎪⎪⎨⎪⎪⎩⎛⎜⎜⎝𝑛+𝑟+𝑘𝑠+1_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑘+𝑟−𝑛−1≤𝑎≤𝑟}⎞⎟⎟⎠⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^{𝑠−𝑐𝑛+𝑎+1−𝑘−𝑟}_∞𝑡=0_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑡}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐−_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘+1}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭𝐵_𝑘+1(𝑥)2⎛⎜⎜⎝𝑛+𝑛−𝑟+1𝑠⎞⎟⎟⎠𝐵^𝑛−1+(𝑥)+⎛⎜⎜⎝𝑛+𝑛𝑟+𝑠⎞⎟⎟⎠𝐵_𝑛(𝑥)_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠𝑖+𝑡=𝑛1}_𝑙1=0⋯^𝑖𝑟_{𝑙𝑟𝑗=0}¹_𝑝1=0⋯^𝑗𝑠_𝑝_𝑠=0⎧⎪⎨⎪⎩⎛⎜⎜⎝𝑖₁𝑙₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑖_𝑟𝑙_𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑗₁𝑝₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑗_𝑠𝑝_𝑠×𝐵_{𝑖1−𝑙1}⎞⎟⎟⎠⋯𝐵_{𝑖𝑟−𝑙𝑟}𝐸_{𝑗1−𝑝1}⋯𝐸_{𝑗𝑠−𝑝𝑠}×1𝑙₁+⋯+𝑙_𝑟+𝑝₁+⋯𝑝_𝑠+𝑡+1⎫⎪⎬⎪⎭.(2.11)

Consider the following polynomial of degree 𝑛: 𝑝(𝑥)=^∞_{𝑖1+⋯+𝑖𝑟+}_{𝑗1+⋯+𝑗𝑠=𝑛}1𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝐵^𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥).(2.12) Then, from (2.12), one has 𝑝^(𝑘)(𝑥)=(𝑟+𝑠)^𝑘_{𝑖1+⋯+𝑖𝑟+𝑗}_{1+⋯+𝑗𝑠=𝑛−𝑘}𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!.(2.13) By (1.4) and (2.13), one gets, for 𝑘=1,2,…,𝑛, 𝑎_𝑘=1𝑘!𝑝^(𝑘−1)=(𝑟+𝑠)_𝑘−1(1)−𝑝^(𝑘−1)(0)𝑘!_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛−𝑘+1}𝐵^𝑖1(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠=(𝑟+𝑠)_𝑘−1𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑘!⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑘+𝑟−𝑛−1≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯}_{+𝑗𝑐=𝑛+𝑎+1−𝑘−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐−_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘+1}𝑖₁!𝑖₂!⋯𝑖_𝑎!𝑗₁!⋯𝑗_𝑐!1𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝐵^𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭.(2.14) Now look at 𝑎_𝑛 and 𝑎_𝑛−1: 𝑎_𝑛=(𝑟+𝑠)^𝑛−1𝑛!_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=1}𝐵^𝑖1(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠=(𝑟+𝑠)_𝑛−1𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑎𝑛!12(𝑟+𝑠)−−12(𝑟+𝑠)=(𝑟+𝑛!_𝑛−1𝑠)^𝑛,=(𝑟+𝑠)^𝑛−2(𝑛−1)!_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=2}𝐵^𝑖1(1)⋯𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠=(𝑟+𝑠)_𝑛−2𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!(𝑛−1)!⎧⎪⎨⎪⎩1261𝑟+2121⎛⎜⎜⎝𝑟+2𝑠⎞⎟⎟⎠−2161𝑟−12−12⎛⎜⎜⎝𝑟+2𝑠⎞⎟⎟⎠⎫⎪⎬⎪⎭=0.(2.15)

It is easy to show that 𝑎₀=¹⁰𝑝(𝑡)𝑑𝑡=_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛}1×_𝑖1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!_𝑙₁₌₀⋯^𝑖𝑟_{𝑙𝑟𝑗=0}¹_𝑝1=0⋯^𝑗𝑠_𝑝_𝑠=0⎧⎪⎨⎪⎩𝐵_{𝑖1−𝑙1}⋯𝐵_{𝑖𝑟−𝑙𝑟}𝐸_{𝑗1−𝑝1}𝐸_{𝑗𝑠−𝑝𝑠}𝑙₁+⋯+𝑙_𝑟+𝑝₁+⋯𝑝_𝑠+1⎛⎜⎜⎝𝑙𝑖¹₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑖_𝑟𝑙_𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑗₁𝑝₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑗_𝑠𝑝_𝑠⎞⎟⎟⎠⎫⎪⎬⎪⎭.(2.16) Therefore, by (1.3), (2.14), and (2.15), one obtains the following theorem.

Theorem 2.3. For 𝑛∈ℕ with 𝑛≥2, one has _{𝑖1+⋯+𝑖𝑟+}_{𝑗1+⋯+𝑗𝑠=𝑛}1𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝐵^𝑖1=(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)_𝑛−2_𝑘=1(𝑟+𝑠)^𝑘−1𝑘!⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑘+𝑟−𝑛−1≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^𝑠−𝑐×_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑛+𝑎+1−𝑘−𝑟}𝐵_𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐−_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘+1}𝑖₁!𝑖₂!⋯𝑖_𝑎!𝑗₁!⋯𝑗_𝑐!1×𝐵_𝑖1𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭𝐵_𝑘(𝑥)+(𝑟+𝑠)^𝑛𝑛!𝐵^𝑛+(𝑥)_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠𝑖=𝑛1}_𝑙1=0⋯^𝑖𝑟_{𝑙𝑟𝑗=0}¹_𝑝1=0⋯^𝑗𝑠_𝑝_𝑠=0⎛⎜⎜⎝𝑖₁𝑙₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑖_𝑟𝑙_𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑗₁𝑝₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑗_𝑠𝑝_𝑠×𝐵_{𝑖1−𝑙1}⎞⎟⎟⎠⋯𝐵_{𝑖𝑟−𝑙𝑟}𝐸_{𝑗1−𝑝1}𝐸_{𝑗𝑠−𝑝𝑠}𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑙₁+⋯+𝑙_𝑟+𝑝₁+⋯𝑝_𝑠+1.(2.17)

Take the polynomial 𝑝(𝑥) of degree 𝑛 as follows: 𝑝(𝑥)=_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛}1𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!𝐵^𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑥^𝑡.(2.18) Then, from (2.18), one gets 𝑝^(𝑘)(𝑥)=(𝑟+𝑠+1)^𝑘×_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛−𝑘}1𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!𝐵^𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑥^𝑡.(2.19) By (1.4) and (2.19), one gets, for 𝑘=1,…,𝑛, 𝑎_𝑘=1𝑘!𝑝^(𝑘−1)=(𝑟+𝑠+1)(1)_𝑘−1−𝑝^(𝑘−1)(0)𝑘!_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛−𝑘+1}1×𝐵𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!_𝑖1=(𝑟+𝑠+(11))⋯_𝑘−1𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠0^𝑡𝑘!⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑘+𝑟−𝑛−1≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^{𝑠−𝑐𝑛+𝑎+1−𝑘−𝑟}_𝑡=01×(𝑛+𝑎+1−𝑘−𝑟−𝑡)!_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑡}1𝑖₁!𝑖₂!⋯𝑖_𝑎!𝑗₁!⋯𝑗_𝑐!𝐵^𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐−_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘+1}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠⎫⎪⎪⎬⎪⎪⎭.(2.20)

Now look at 𝑎_𝑛 and 𝑎_𝑛−1: 𝑎_𝑛=(𝑟+𝑠+1)^𝑛−1𝑛!_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=1}1×𝐵𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!_𝑖1=(𝑟+𝑠+(11))⋯_𝑛−1𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠0^𝑡=(𝑟+𝑠𝑛+!1)_𝑛−121(𝑟+𝑠)+1−−12(𝑟+𝑠)𝑎𝑛!(𝑟+𝑠+1)=(𝑟+𝑠𝑛!_𝑛−1+1)^𝑛,=(𝑟+𝑠+1)^𝑛−2(𝑛−1)!_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=2}1×𝐵𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!_𝑖1=(𝑟+𝑠+(11))⋯_𝑛−2𝐵_𝑖𝑟(1)𝐸_𝑗1(1)⋯𝐸_𝑗𝑠(1)−𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠0^𝑡=(𝑟+(𝑛𝑠−+11))!_𝑛−2⎧⎪⎨⎪⎩1261𝑟+21+1221⎛⎜⎜⎝𝑟+2𝑠⎞⎟⎟⎠+21(𝑟+𝑠)−2161𝑟−−21−12⎛⎜⎜⎝𝑟+2𝑠⎞⎟⎟⎠⎫⎪⎬⎪⎭=(𝑟+(𝑛𝑠−+11))!_𝑛−1𝑟+𝑠2+12(𝑛−1)!.(2.21) From (2.18), one can derive the following identity: 𝑎₀==¹⁰𝑝(𝑡)𝑑𝑡_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛}1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!¹⁰=𝐵_𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑥^𝑡𝑑𝑡_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠+𝑡=𝑛}1𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!^𝑖1_𝑙1=0⋯^𝑖𝑟_{𝑙𝑟𝑗=0}¹_𝑝1=0×_𝑗𝑠⋯_𝑝_𝑠=0⎛⎜⎜⎝𝑖₁𝑙₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑖_𝑟𝑙_𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑗₁𝑝₁⎞⎟⎟⎠⋯⎛⎜⎜⎝𝑗_𝑠𝑝_𝑠⎞⎟⎟⎠𝐵_{𝑖1−𝑙1}⋯𝐵_{𝑖𝑟−𝑙𝑟}𝐸_{𝑗1−𝑝1}𝐸_{𝑗𝑠−𝑝𝑠}1𝑙₁+⋯+𝑙_𝑟+𝑝₁+⋯𝑝_𝑠+𝑡+1.(2.22) Therefore, by (1.3), (2.20), (2.21), and (2.22), one obtains the following theorem.

Theorem 2.4. For 𝑛∈ℕ with 𝑛≥2, one has _{𝑖1+⋯+𝑖𝑟+}_{𝑗1+⋯+𝑗𝑠+𝑡=𝑛}1𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!𝐵^𝑖1(𝑥)⋯𝐵_𝑖𝑟(𝑥)𝐸_𝑗1(𝑥)⋯𝐸_𝑗𝑠(𝑥)𝑥^𝑡=^𝑛−2_𝑘=1(𝑟+𝑠+1)^𝑘−1𝑘!⎧⎪⎪⎨⎪⎪⎩_{0≤𝑎≤𝑟0≤𝑐≤𝑠𝑘+𝑟−𝑛−1≤𝑎≤𝑟}⎛⎜⎜⎝𝑎𝑟⎞⎟⎟⎠⎛⎜⎜⎝𝑐𝑠⎞⎟⎟⎠(−1)_𝑐2^{𝑠−𝑐𝑛+𝑎+1−𝑘−𝑟}_𝑡=01×(𝑛+𝑎+1−𝑘−𝑟−𝑡)!_{𝑖1+⋯+𝑖𝑎+𝑗1+⋯+𝑗𝑐=𝑡}1𝑖₁!𝑖₂!⋯𝑖_𝑎!𝑗₁!⋯𝑗_𝑐!𝐵^𝑖1⋯𝐵_𝑖𝑎𝐸_𝑗1⋯𝐸_𝑗𝑐−_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠=𝑛−𝑘+1}𝐵_𝑖1⋯𝐵_𝑖𝑟𝐸_𝑗1⋯𝐸_𝑗𝑠𝑖₁!𝑖₂!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!⎫⎪⎪⎬⎪⎪⎭𝐵^𝑘+(𝑟+𝑠+1)_𝑛−1(𝑥)2(𝑛−1)!𝐵^𝑛−1(𝑥)+(𝑟+𝑠+1)^𝑛𝑛!_{𝑖1+⋯+𝑖𝑟+𝑗1+⋯+𝑗𝑠𝑖+𝑡=𝑛}¹_𝑙1=0⋯^𝑖𝑟_{𝑙𝑟𝑗=0}¹_𝑝1=0⋯^𝑗𝑠_𝑝_𝑠=0×^𝑖1𝑙1⋯^𝑖𝑟𝑙_𝑟^𝑗1𝑝1⋯^𝑗𝑠𝑝_𝑠𝑖₁!⋯𝑖_𝑟!𝑗₁!⋯𝑗_𝑠!𝑡!𝐵^{𝑖1−𝑙1}⋯𝐵_{𝑖𝑟−𝑙𝑟}𝐸_{𝑗1−𝑝1}𝐸_{𝑗𝑠−𝑝𝑠}1𝑙₁+⋯+𝑙_𝑟+𝑝₁+⋯𝑝_𝑠+𝑡+1.(2.23)

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.