Advances in Difference Equations
VolumeΒ 2009Β (2009), Article IDΒ 273545, 9 pages
doi:10.1155/2009/273545
Research Article

On the Identities of Symmetry for the 𝜁 -Euler Polynomials of Higher Order

Taekyun Kim,1Β Kyoung Ho Park,2Β and Kyung-won Hwang3

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
2Department of Mathematics, Sogang University, Seoul 121-742, South Korea
3Department of General Education, Kookmin University, Seoul 139-702, South Korea

Received 19 February 2009; Revised 31 May 2009; Accepted 18 June 2009

Academic Editor: AgacikΒ Zafer

Copyright Β© 2009 Taekyun Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this paper is to investigate several further interesting properties of symmetry for the multivariate 𝑝 -adic fermionic integral on β„€ 𝑝 . From these symmetries, we can derive some recurrence identities for the 𝜁 -Euler polynomials of higher order, which are closely related to the Frobenius-Euler polynomials of higher order. By using our identities of symmetry for the 𝜁 -Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher order.

1. Introduction/Definition

Let 𝑝 be a fixed odd prime number. Throughout this paper, β„€ 𝑝 , β„š 𝑝 , β„‚ , and β„‚ 𝑝 will, respectively, denote the ring of 𝑝 -adic rational integer, the field of 𝑝 -adic rational numbers, the complex number field, and the completion of algebraic closure of β„š 𝑝 . Let 𝑣 𝑝 be the normalized exponential valuation of β„‚ 𝑝 with | 𝑝 | 𝑝 = 𝑝 βˆ’ 𝑣 𝑝 ( 𝑝 ) = 𝑝 βˆ’ 1 . Let π‘ˆ 𝐷 ( β„€ 𝑝 ) be the space of uniformly differentiable functions on β„€ 𝑝 . For 𝑓 ∈ π‘ˆ 𝐷 ( β„€ 𝑝 ) , π‘ž ∈ β„‚ 𝑝 with | 1 βˆ’ π‘ž | 𝑝 < 1 , the fermionic 𝑝 -adic π‘ž -integral on β„€ 𝑝 is defined as 𝐼 βˆ’ π‘ž ( ξ€œ 𝑓 ) = β„€ 𝑝 𝑓 ( π‘₯ ) 𝑑 πœ‡ βˆ’ π‘ž ( π‘₯ ) = l i m 𝑁 β†’ ∞ 1 + π‘ž 1 + π‘ž 𝑝 𝑁 𝑝 𝑁 βˆ’ 1  π‘₯ = 0 𝑓 ( π‘₯ ) ( βˆ’ π‘ž ) x ( 1 . 1 ) (see [1]). Let us define the fermionic 𝑝 -adic invariant integral on β„€ 𝑝 as follows: 𝐼 βˆ’ 1 ( 𝑓 ) = l i m π‘ž β†’ 1 𝐼 βˆ’ π‘ž ( ξ€œ 𝑓 ) = β„€ 𝑝 𝑓 ( π‘₯ ) 𝑑 πœ‡ βˆ’ 1 ( π‘₯ ) ( 1 . 2 ) (see [1–8]). From (1.2), we have 𝐼 βˆ’ 1 ξ€· 𝑓 1 ξ€Έ + 𝐼 βˆ’ 1 ( 𝑓 ) = 2 𝑓 ( 0 ) ( 1 . 3 ) (see [9, 10]), where 𝑓 1 ( π‘₯ ) = 𝑓 ( π‘₯ + 1 ) . For 𝜁 ∈ β„‚ 𝑝 with | 1 βˆ’ 𝜁 | 𝑝 < 1 , let 𝑓 ( π‘₯ ) = 𝑒 π‘₯ 𝑑 𝜁 π‘₯ . Then, we define the 𝜁 -Euler numbers as follows: ξ€œ β„€ 𝑝 𝜁 π‘₯ 𝑒 π‘₯ 𝑑 𝑑 πœ‡ βˆ’ 1 ( 2 π‘₯ ) = 𝜁 𝑒 𝑑 = + 1 ∞  𝑛 = 0 𝐸 𝑛 , 𝜁 𝑑 𝑛 , 𝑛 ! ( 1 . 4 ) where 𝐸 𝑛 , 𝜁 are called the 𝜁 -Euler numbers. We can show that 2 𝜁 𝑒 𝑑 = + 1 1 + 𝜁 βˆ’ 1 𝑒 𝑑 + 𝜁 βˆ’ 1 β‹… 2 = 2 1 + 𝜁 1 + 𝜁 ∞  𝑛 = 0 𝐻 𝑛 ξ€· βˆ’ 𝜁 βˆ’ 1 ξ€Έ 𝑑 𝑛 , 𝑛 ! ( 1 . 5 ) where 𝐻 𝑛 ( βˆ’ 𝜁 βˆ’ 1 ) are the Frobenius-Euler numbers. By comparing the coefficients on both sides of (1.4) and (1.5), we see that 𝐸 𝑛 , 𝜁 = 2 𝐻 1 + 𝜁 𝑛 ξ€· βˆ’ 𝜁 βˆ’ 1 ξ€Έ . ( 1 . 6 ) Now, we also define the 𝜁 -Euler polynomials as follows: 2 𝜁 𝑒 𝑑 𝑒 + 1 π‘₯ 𝑑 = ∞  𝑛 = 0 𝐸 𝑛 , 𝜁 ( 𝑑 π‘₯ ) 𝑛 . 𝑛 ! ( 1 . 7 ) In the viewpoint of (1.5), we can show that 2 𝜁 𝑒 𝑑 𝑒 + 1 π‘₯ 𝑑 = 𝑒 π‘₯ 𝑑 1 + 𝜁 βˆ’ 1 𝑒 𝑑 + 𝜁 βˆ’ 1 β‹… 2 = 2 1 + 𝜁 1 + 𝜁 ∞  𝑛 = 0 𝐻 𝑛 ξ€· βˆ’ 𝜁 βˆ’ 1 ξ€Έ 𝑑 , π‘₯ 𝑛 , 𝑛 ! ( 1 . 8 ) where 𝐻 𝑛 ( βˆ’ 𝜁 βˆ’ 1 , π‘₯ ) are the 𝑛 th Frobenius-Euler polynomials. From (1.7) and (1.8), we note that 𝐸 𝑛 , 𝜁 2 ( π‘₯ ) = 𝐻 1 + 𝜁 𝑛 ξ€· βˆ’ 𝜁 βˆ’ 1 ξ€Έ , π‘₯ ( 1 . 9 ) (cf. [1–8, 11–18]). For each positive integer π‘˜ , let 𝑇 π‘˜ , 𝜁 βˆ‘ ( 𝑛 ) = 𝑛 β„“ = 0 ( βˆ’ 1 ) β„“ 𝜁 β„“ β„“ π‘˜ . Then we have

∞  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑑 ( 𝑛 ) π‘˜ = π‘˜ ! ∞  π‘˜ = 0  𝑛  β„“ = 0 ( βˆ’ 1 ) β„“ β„“ π‘˜ 𝜁 β„“ ξƒͺ 𝑑 π‘˜ = π‘˜ ! 𝑛  β„“ = 0 ( βˆ’ 1 ) β„“ 𝜁 β„“ 𝑒 β„“ 𝑑 = 1 + ( βˆ’ 1 ) 𝑛 + 1 𝑒 ( 𝑛 + 1 ) 𝑑 𝜁 𝑒 𝑑 . + 1 ( 1 . 1 0 ) The 𝜁 -Euler polynomials of order π‘˜ , denoted 𝐸 ( π‘˜ ) 𝑛 , 𝜁 ( π‘₯ ) , are defined as 𝑒 π‘₯ 𝑑 ξ‚΅ 2 𝜁 𝑒 𝑑 ξ‚Ά + 1 π‘˜ = ξ‚΅ 2 𝜁 𝑒 𝑑 ξ‚Ά ξ‚΅ 2 + 1 Γ— β‹― Γ— 𝜁 𝑒 𝑑 ξ‚Ά 𝑒 + 1 π‘₯ 𝑑 = ∞  𝑛 = 0 𝐸 ( π‘˜ ) 𝑛 , 𝜁 ( 𝑑 π‘₯ ) 𝑛 . 𝑛 ! ( 1 . 1 1 ) Then the values of 𝐸 ( π‘˜ ) 𝑛 , 𝜁 ( π‘₯ ) at π‘₯ = 0 are called the 𝜁 -Euler numbers of order π‘˜ . When π‘˜ = 1 , the polynomials or numbers are called the 𝜁 -Euler polynomials or numbers. The purpose of this paper is to investigate some properties of symmetry for the multivariate 𝑝 -adic fermionic integral on β„€ 𝑝 . From the properties of symmetry for the multivariate 𝑝 -adic fermionic integral on β„€ 𝑝 , we derive some identities of symmetry for the 𝜁 -Euler polynomials of higher order. By using our identities of symmetry for the 𝜁 -Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher order.

2. On the Symmetry for the 𝜻 -Euler Polynomials of Higher Order

Let 𝑀 1 , 𝑀 2 ∈ β„• with 𝑀 1 ≑ 1 (mod 2) and 𝑀 2 ≑ 1 ( m o d 2 ) . Then we set 𝑅 ( π‘š ) ξ€· w 1 , 𝑀 2 ξ€Έ = ∫ β„€ π‘š 𝑝 𝑒 𝑀 1 ( π‘₯ 1 + π‘₯ 2 + β‹― + π‘₯ π‘š + 𝑀 2 π‘₯ ) 𝑑 𝜁 𝑀 1 π‘₯ 1 + β‹― + 𝑀 1 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ∫ β„€ 𝑝 𝜁 𝑀 1 𝑀 2 π‘₯ 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 𝑑 πœ‡ βˆ’ 1 Γ— ξ€œ ( π‘₯ ) β„€ π‘š 𝑝 𝑒 𝑀 2 ( π‘₯ 1 + π‘₯ 2 + β‹― + π‘₯ π‘š + 𝑀 1 𝑦 ) 𝑑 𝜁 𝑀 2 π‘₯ 1 + β‹― + 𝑀 2 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ , ( 2 . 1 ) where ξ€œ β„€ π‘š 𝑝 𝑓 ξ€· π‘₯ 1 , … , π‘₯ π‘š ξ€Έ 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ = ξ€œ β„€ 𝑝 β‹― ξ€œ β„€ 𝑝 𝑓 ξ€· π‘₯ 1 , … , π‘₯ π‘š ξ€Έ 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ . ( 2 . 2 ) Thus, we note that this expression for 𝑅 ( π‘š ) ( 𝑀 1 , 𝑀 2 ) is symmetry in 𝑀 1 and 𝑀 2 . From (2.1), we have 𝑅 ( π‘š ) ξ€· 𝑀 1 , 𝑀 2 ξ€Έ =  ξ€œ β„€ π‘š 𝑝 𝑒 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 𝜁 𝑀 1 π‘₯ 1 + β‹― + 𝑀 1 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 Γ— βŽ› ⎜ ⎜ ⎝ ∫ β„€ 𝑝 𝑒 𝑀 2 π‘₯ π‘š 𝑑 𝜁 𝑀 2 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ∫ β„€ 𝑝 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 𝜁 𝑀 1 𝑀 2 π‘₯ 𝑑 πœ‡ βˆ’ 1 ⎞ ⎟ ⎟ ⎠ Γ—  ξ€œ ( π‘₯ ) β„€ 𝑝 π‘š βˆ’ 1 𝑒 𝑀 2 ξ€· π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 ξ€Έ 𝑑 𝜁 𝑀 2 π‘₯ 1 + β‹― + 𝑀 2 π‘₯ π‘š βˆ’ 1 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š βˆ’ 1 ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 𝑦 𝑑 . ( 2 . 3 ) We can show that ∫ β„€ 𝑝 𝑒 π‘₯ 𝑑 𝜁 π‘₯ 𝑑 πœ‡ βˆ’ 1 ( π‘₯ ) ∫ β„€ 𝑝 𝑒 𝑀 1 π‘₯ 𝑑 𝜁 𝑀 1 π‘₯ 𝑑 πœ‡ βˆ’ 1 = ( π‘₯ ) 𝑀 1 βˆ’ 1  β„“ = 0 ( βˆ’ 1 ) β„“ 𝜁 β„“ 𝑒 β„“ 𝑑 = ∞  π‘˜ = 0 ξ€· 𝑇 π‘˜ , 𝜁 ξ€· 𝑀 1 𝑑 βˆ’ 1 ξ€Έ ξ€Έ π‘˜ . π‘˜ ! ( 2 . 4 ) By (1.4) and (1.11), we see that  ξ€œ β„€ π‘š 𝑝 𝑒 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 𝜁 𝑀 1 π‘₯ 1 + β‹― + 𝑀 1 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 = ξ‚΅ 2 𝜁 𝑀 1 𝑒 𝑀 1 𝑑 ξ‚Ά + 1 π‘š 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 = ∞  𝑛 = 0 𝐸 ( π‘š ) 𝑛 , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ 𝑀 𝑛 1 𝑑 𝑛 . 𝑛 ! ( 2 . 5 ) Thus, we have 𝐸 ( π‘š ) 𝑛 , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ = 𝑛  β„“ = 0 ξ‚΅ 𝑛 β„“ ξ‚Ά 𝐸 ( π‘š ) β„“ , 𝜁 𝑀 1 𝑀 2 𝑛 βˆ’ β„“ π‘₯ 𝑛 βˆ’ β„“ . ( 2 . 6 ) From (2.3), (2.4), and (2.5), we can derive 𝑅 ( π‘š ) ξ€· 𝑀 1 , 𝑀 2 ξ€Έ =  ∞  β„“ = 0 𝐸 ( π‘š ) β„“ , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ 𝑀 β„“ 1 𝑑 β„“ β„“ ! ξƒͺ  ∞  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 ξ€Έ 𝑀 βˆ’ 1 π‘˜ 2 𝑑 π‘˜ ! π‘˜ ξƒͺ  ∞  𝑖 = 0 𝐸 ( π‘š βˆ’ 1 ) 𝑖 , 𝜁 𝑀 2 ξ€· 𝑀 1 𝑦 ξ€Έ 𝑀 𝑖 2 𝑑 𝑖 ! 𝑖 ξƒͺ =  ∞  β„“ = 0 𝐸 ( π‘š ) β„“ , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ 𝑀 β„“ 1 𝑑 β„“ β„“ ! ξƒͺ  ∞  𝑗 = 0  𝑗  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 ξ€Έ 𝑀 βˆ’ 1 π‘˜ 2 𝑀 2 𝑗 βˆ’ π‘˜ 𝐸 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ ξ€· 𝑀 1 𝑦 ξ€Έ ξƒͺ 𝑑 π‘˜ ! ( 𝑗 βˆ’ π‘˜ ) ! 𝑗 ! 𝑗 ξƒͺ =  𝑗 ! ∞  β„“ = 0 𝐸 ( π‘š ) β„“ , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ 𝑀 β„“ 1 𝑑 β„“ β„“ ! ξƒͺ  ∞  𝑗 = 0  𝑗  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 ξ€Έ ξ‚΅ 𝑗 π‘˜ ξ‚Ά 𝐸 βˆ’ 1 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 𝑦 ξ€Έ ξƒͺ 𝑀 𝑗 2 ξƒͺ 𝑑 𝑗 = 𝑗 ! ∞  𝑛 = 0  𝑛  𝑗 = 0  𝑗  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 ξ€Έ ξ‚΅ 𝑗 π‘˜ ξ‚Ά 𝐸 βˆ’ 1 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 𝑦 ξ€Έ ξƒͺ 𝑀 𝑗 2 𝑀 1 𝑛 βˆ’ 𝑗 𝐸 ( 𝑛 βˆ’ 𝑗 ) ! 𝑗 ! ( π‘š ) 𝑛 βˆ’ 𝑗 , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ ξƒͺ 𝑑 𝑛 ! 𝑛 = 𝑛 ! ∞  𝑛 = 0  𝑛  𝑗 = 0 ξ‚΅ 𝑛 𝑗 ξ‚Ά 𝑀 𝑗 2 𝑀 1 𝑛 βˆ’ 𝑗 𝐸 ( π‘š ) 𝑛 βˆ’ 𝑗 , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ 𝑗  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 ξ€Έ ξ‚΅ 𝑗 π‘˜ ξ‚Ά 𝐸 βˆ’ 1 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 𝑦 ξ€Έ ξƒͺ 𝑑 𝑛 . 𝑛 ! ( 2 . 7 ) By the same method, we also see that 𝑅 ( π‘š ) ξ€· 𝑀 1 , 𝑀 2 ξ€Έ =  ξ€œ β„€ π‘š 𝑝 𝑒 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 𝜁 𝑀 2 π‘₯ 1 + β‹― + 𝑀 2 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 Γ— βŽ› ⎜ ⎜ ⎝ ∫ β„€ 𝑝 𝑒 𝑀 1 π‘₯ π‘š 𝑑 𝜁 𝑀 1 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ∫ β„€ 𝑝 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 𝜁 𝑀 1 𝑀 2 π‘₯ 𝑑 πœ‡ βˆ’ 1 ⎞ ⎟ ⎟ ⎠ Γ—  ξ€œ ( π‘₯ ) β„€ π‘š βˆ’ 1 p 𝑒 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 ) 𝑑 𝜁 𝑀 1 π‘₯ 1 + β‹― + 𝑀 1 π‘₯ π‘š βˆ’ 1 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š βˆ’ 1 ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 𝑦 𝑑 =  ∞  β„“ = 0 𝐸 ( π‘š ) β„“ , 𝜁 𝑀 2 ξ€· 𝑀 1 π‘₯ ξ€Έ 𝑀 β„“ 2 𝑑 β„“ β„“ ! ξƒͺ  ∞  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 ξ€Έ 𝑀 βˆ’ 1 π‘˜ 1 𝑑 π‘˜ π‘˜ ! ξƒͺ  ∞  𝑖 = 0 𝐸 ( π‘š βˆ’ 1 ) 𝑖 , 𝜁 𝑀 1 ξ€· 𝑀 2 𝑦 ξ€Έ 𝑀 𝑖 1 𝑑 𝑖 ξƒͺ =  𝑖 ! ∞  β„“ = 0 𝐸 ( π‘š ) β„“ , 𝜁 𝑀 2 ξ€· 𝑀 1 π‘₯ ξ€Έ 𝑀 β„“ 2 𝑑 β„“ β„“ ! ξƒͺ  ∞  𝑗 = 0  𝑗  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 ξ€Έ βˆ’ 1 𝐸 π‘˜ ! ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ ξ€· 𝑀 2 𝑦 ξ€Έ ( ξƒͺ 𝑀 𝑗 βˆ’ π‘˜ ) ! 𝑗 1 𝑑 𝑗 ξƒͺ =  ∞  β„“ = 0 𝐸 ( π‘š ) β„“ , 𝜁 𝑀 2 ξ€· 𝑀 1 π‘₯ ξ€Έ 𝑀 β„“ 2 𝑑 β„“ β„“ ! ξƒͺ  ∞  𝑗 = 0  𝑗  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 ξ€Έ 𝐸 βˆ’ 1 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ ξ€· 𝑀 2 𝑦 ξ€Έ ξƒͺ 𝑀 π‘˜ ! ( 𝑗 βˆ’ π‘˜ ) ! 𝑗 ! 𝑗 1 𝑑 𝑗 ξƒͺ =  𝑗 ! ∞  β„“ = 0 𝐸 ( π‘š ) β„“ , 𝜁 𝑀 2 ξ€· 𝑀 1 π‘₯ ξ€Έ 𝑀 β„“ 2 𝑑 β„“ β„“ ! ξƒͺ  ∞  𝑗 = 0  𝑗  π‘˜ = 0 ξ‚΅ 𝑗 π‘˜ ξ‚Ά 𝑇 π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 ξ€Έ 𝐸 βˆ’ 1 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 𝑦 ξ€Έ ξƒͺ 𝑀 𝑗 1 𝑑 𝑗 ξƒͺ = 𝑗 ! ∞  𝑛 = 0  𝑛  𝑗 = 0  𝑗  π‘˜ = 0 ξ‚΅ 𝑗 π‘˜ ξ‚Ά 𝑇 π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 ξ€Έ 𝐸 βˆ’ 1 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 𝑦 ξ€Έ ξƒͺ 𝑀 𝑗 1 𝑀 2 𝑛 βˆ’ 𝑗 𝐸 𝑗 ! ( 𝑛 βˆ’ 𝑗 ) ! ( π‘š ) 𝑛 βˆ’ 𝑗 , 𝜁 𝑀 2 ξ€· 𝑀 1 π‘₯ ξ€Έ ξƒͺ 𝑑 𝑛 ! 𝑛 = 𝑛 ! ∞  𝑛 = 0  𝑛  𝑗 = 0 ξ‚΅ 𝑛 𝑗 ξ‚Ά 𝑀 𝑗 1 𝑀 2 𝑛 βˆ’ 𝑗 𝐸 ( π‘š ) 𝑛 βˆ’ 𝑗 , 𝜁 𝑀 2 ξ€· 𝑀 1 π‘₯ ξ€Έ 𝑗  π‘˜ = 0 ξ‚΅ 𝑗 π‘˜ ξ‚Ά 𝑇 π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 ξ€Έ 𝐸 βˆ’ 1 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 𝑦 ξ€Έ ξƒͺ 𝑑 𝑛 . 𝑛 ! ( 2 . 8 ) By comparing the coefficients on both sides of (2.7) and (2.8), we obtain the following.

Theorem 2.1. For 𝑀 1 , 𝑀 2 ∈ β„•   with   𝑀 1 ≑ 1 ( m o d 2 ) , 𝑀 2 ≑ 1 ( m o d 2 ) , and   𝑛 β‰₯ 0 , π‘š β‰₯ 1 , one has 𝑛  𝑗 = 0 ξ‚΅ 𝑛 𝑗 ξ‚Ά 𝑀 𝑗 2 𝑀 1 𝑛 βˆ’ 𝑗 𝐸 ( π‘š ) 𝑛 βˆ’ 𝑗 , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ 𝑗  π‘˜ = 0 𝑇 π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 ξ€Έ ξ‚΅ 𝑗 π‘˜ ξ‚Ά 𝐸 βˆ’ 1 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 𝑦 ξ€Έ = 𝑛  𝑗 = 0 ξ‚΅ 𝑛 𝑗 ξ‚Ά 𝑀 𝑗 1 𝑀 2 𝑛 βˆ’ 𝑗 𝐸 ( π‘š ) 𝑛 βˆ’ 𝑗 , 𝜁 𝑀 2 ξ€· 𝑀 1 π‘₯ ξ€Έ 𝑗  π‘˜ = 0 ξ‚΅ 𝑗 π‘˜ ξ‚Ά 𝑇 π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 ξ€Έ 𝐸 βˆ’ 1 ( π‘š βˆ’ 1 ) 𝑗 βˆ’ π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 𝑦 ξ€Έ . ( 2 . 9 ) Let 𝑦 = 0 and π‘š = 1 in (2.9). Then we have 𝑛  𝑗 = 0 ξ‚΅ 𝑛 𝑗 ξ‚Ά 𝑀 1 𝑛 βˆ’ 𝑗 𝑀 𝑗 2 𝐸 𝑛 βˆ’ 𝑗 , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ 𝑇 π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 ξ€Έ = βˆ’ 1 𝑛  𝑗 = 0 ξ‚΅ 𝑛 𝑗 ξ‚Ά 𝑀 𝑗 1 𝑀 2 𝑛 βˆ’ 𝑗 𝐸 𝑛 βˆ’ 𝑗 , 𝜁 𝑀 2 ξ€· 𝑀 1 π‘₯ ξ€Έ 𝑇 π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 ξ€Έ . βˆ’ 1 ( 2 . 1 0 ) From (2.10), we note that 𝑛  𝑖 = 0 ξ‚΅ 𝑛 𝑖 ξ‚Ά 𝑀 𝑖 1 𝑀 2 𝑛 βˆ’ 𝑖 𝐸 𝑖 , 𝜁 𝑀 1 ξ€· 𝑀 2 π‘₯ ξ€Έ 𝑇 𝑛 βˆ’ 𝑖 , 𝜁 𝑀 2 ξ€· 𝑀 1 ξ€Έ = βˆ’ 1 𝑛  𝑖 = 0 ξ‚΅ 𝑛 𝑖 ξ‚Ά 𝑀 1 𝑛 βˆ’ 𝑖 𝑀 𝑖 2 𝐸 𝑖 , 𝜁 𝑀 2 ξ€· 𝑀 1 π‘₯ ξ€Έ 𝑇 𝑛 βˆ’ 𝑖 , 𝜁 w 1 ξ€· 𝑀 2 ξ€Έ . βˆ’ 1 ( 2 . 1 1 ) If we take 𝑀 2 = 1 in (2.11), then we have 𝐸 𝑛 , 𝜁 ξ€· 𝑀 1 π‘₯ ξ€Έ = 𝑛  𝑖 = 0 ξ‚΅ 𝑛 𝑖 ξ‚Ά 𝑀 𝑖 1 𝐸 𝑖 , 𝜁 𝑀 1 ( π‘₯ ) 𝑇 𝑛 βˆ’ 𝑖 , 𝜁 ξ€· 𝑀 1 ξ€Έ . βˆ’ 1 ( 2 . 1 2 ) From (2.3), we note that 𝑅 ( π‘š ) ξ€· 𝑀 1 , 𝑀 2 ξ€Έ =  ξ€œ β„€ π‘š 𝑝 𝑒 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 𝜁 𝑀 1 π‘₯ 1 + β‹― + 𝑀 1 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 Γ— βŽ› ⎜ ⎜ ⎝ ∫ β„€ 𝑝 𝑒 𝑀 2 π‘₯ π‘š 𝑑 𝜁 𝑀 2 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ∫ β„€ 𝑝 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 𝜁 𝑀 1 𝑀 2 π‘₯ 𝑑 πœ‡ βˆ’ 1 ⎞ ⎟ ⎟ ⎠ Γ—  ξ€œ ( π‘₯ ) β„€ 𝑝 π‘š βˆ’ 1 𝑒 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 ) 𝑑 𝜁 𝑀 2 π‘₯ 1 + β‹― + 𝑀 2 π‘₯ π‘š βˆ’ 1 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š βˆ’ 1 ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 𝑦 𝑑 =  ξ€œ β„€ π‘š 𝑝 𝑒 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 𝜁 𝑀 1 π‘₯ 1 + β‹― + 𝑀 1 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 Γ— βŽ› ⎜ ⎜ ⎝ 𝑀 1 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝑒 𝑀 2 𝑖 𝑑 𝜁 𝑀 2 𝑖 ⎞ ⎟ ⎟ ⎠ Γ—  ξ€œ β„€ 𝑝 π‘š βˆ’ 1 𝑒 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 ) 𝑑 𝜁 𝑀 2 π‘₯ 1 + β‹― + 𝑀 2 π‘₯ π‘š βˆ’ 1 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š βˆ’ 1 ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 𝑦 𝑑 = βŽ› ⎜ ⎜ ⎝ 𝑀 1 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝜁 𝑀 2 𝑖 ξ€œ β„€ π‘š 𝑝 𝑒 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š + ( 𝑀 2 / 𝑀 1 ) 𝑖 + 𝑀 2 π‘₯ ) 𝑑 𝜁 𝑀 1 π‘₯ 1 + β‹― + 𝑀 1 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ⎞ ⎟ ⎟ ⎠ Γ—  ξ€œ β„€ 𝑝 π‘š βˆ’ 1 𝑒 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 + 𝑀 1 𝑦 ) 𝑑 𝜁 𝑀 2 π‘₯ 1 + β‹― + 𝑀 2 π‘₯ π‘š βˆ’ 1 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š βˆ’ 1 ξ€Έ ξƒͺ = βŽ› ⎜ ⎜ ⎝ 𝑀 1 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝜁 𝑀 2 𝑖 ∞  π‘˜ = 0 𝐸 ( π‘š ) π‘˜ , 𝜁 𝑀 1 ξ‚΅ 𝑀 2 𝑀 1 𝑖 + 𝑀 2 π‘₯ ξ‚Ά 𝑀 π‘˜ 1 𝑑 π‘˜ ⎞ ⎟ ⎟ ⎠  π‘˜ ! ∞  β„“ = 0 𝐸 ( π‘š βˆ’ 1 ) β„“ , 𝜁 𝑀 2 ξ€· 𝑀 1 𝑦 ξ€Έ 𝑀 β„“ 2 𝑑 β„“ ξƒͺ = β„“ ! ∞  𝑛 = 0 βŽ› ⎜ ⎜ ⎝ 𝑛  π‘˜ = 0 βŽ› ⎜ ⎜ ⎝ 𝑀 1 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝜁 𝑀 2 𝑖 𝐸 ( π‘š ) π‘˜ , 𝜁 𝑀 1 ξ‚΅ 𝑀 2 𝑀 π‘₯ + 2 𝑀 1 𝑖 ξ‚Ά ⎞ ⎟ ⎟ ⎠ 𝑀 π‘˜ 1 𝐸 π‘˜ ! ( π‘š βˆ’ 1 ) 𝑛 βˆ’ π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 𝑦 ξ€Έ 𝑀 2 𝑛 βˆ’ π‘˜ ⎞ ⎟ ⎟ ⎠ 𝑑 ( 𝑛 βˆ’ π‘˜ ) ! 𝑛 ! 𝑛 = 𝑛 ! ∞  𝑛 = 0 βŽ› ⎜ ⎜ ⎝ 𝑛  π‘˜ = 0 ξ‚΅ 𝑛 π‘˜ ξ‚Ά 𝑀 π‘˜ 1 𝑀 2 𝑛 βˆ’ π‘˜ 𝐸 ( π‘š βˆ’ 1 ) 𝑛 βˆ’ π‘˜ , 𝜁 𝑀 2 ξ€· 𝑀 1 𝑦 ξ€Έ 𝑀 1 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝜁 𝑀 2 𝑖 𝐸 ( π‘š ) π‘˜ , 𝜁 𝑀 1 ξ‚΅ 𝑀 2 𝑀 π‘₯ + 2 𝑀 1 𝑖 ξ‚Ά ⎞ ⎟ ⎟ ⎠ 𝑑 𝑛 . 𝑛 ! ( 2 . 1 3 ) By the symmetric property of 𝑅 ( π‘š ) ( 𝑀 1 , 𝑀 2 ) in 𝑀 1 , 𝑀 2 , we also see that

𝑅 ( π‘š ) ξ€· 𝑀 1 , 𝑀 2 ξ€Έ =  ξ€œ β„€ π‘š 𝑝 𝑒 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 𝜁 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 Γ— βŽ› ⎜ ⎜ ⎝ ∫ β„€ 𝑝 𝑒 𝑀 1 π‘₯ π‘š 𝑑 𝜁 𝑀 1 π‘₯ π‘š 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ∫ β„€ 𝑝 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 𝜁 𝑀 1 𝑀 2 π‘₯ 𝑑 πœ‡ βˆ’ 1 ⎞ ⎟ ⎟ ⎠ Γ—  ξ€œ ( π‘₯ ) β„€ 𝑝 π‘š βˆ’ 1 𝑒 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 ) 𝑑 𝜁 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 ) 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š βˆ’ 1 ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 𝑦 𝑑 =  ξ€œ β„€ π‘š 𝑝 𝑒 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 𝜁 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ξƒͺ 𝑒 𝑀 1 𝑀 2 π‘₯ 𝑑 Γ— βŽ› ⎜ ⎜ ⎝ 𝑀 2 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝑒 𝑀 1 𝑖 𝑑 𝜁 𝑀 1 𝑖 ⎞ ⎟ ⎟ ⎠ Γ—  ξ€œ β„€ 𝑝 π‘š βˆ’ 1 𝑒 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 + 𝑀 2 𝑦 ) 𝑑 𝜁 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 ) 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š βˆ’ 1 ξ€Έ ξƒͺ = 𝑀 2 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝜁 𝑀 1 𝑖  ξ€œ β„€ π‘š 𝑝 𝑒 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š + ( 𝑀 1 / 𝑀 2 ) 𝑖 + 𝑀 1 π‘₯ ) 𝑑 𝜁 𝑀 2 ( π‘₯ 1 + β‹― + π‘₯ π‘š ) 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š ξ€Έ ξƒͺ Γ—  ξ€œ β„€ 𝑝 π‘š βˆ’ 1 𝑒 𝑀 1 ( π‘₯ 1 + β‹― + π‘₯ π‘š βˆ’ 1 + 𝑀 2 𝑦 ) 𝑑 𝜁 𝑀 1 π‘₯ 1 + β‹― + 𝑀 1 π‘₯ π‘š βˆ’ 1 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ 1 ξ€Έ β‹― 𝑑 πœ‡ βˆ’ 1 ξ€· π‘₯ π‘š βˆ’ 1 ξ€Έ ξƒͺ = βŽ› ⎜ ⎜ ⎝ 𝑀 2 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝜁 𝑀 1 𝑖 ∞  π‘˜ = 0 𝐸 ( π‘š ) π‘˜ , 𝜁 𝑀 2 ξ‚΅ 𝑀 1 𝑀 2 𝑖 + 𝑀 1 π‘₯ ξ‚Ά 𝑀 π‘˜ 2 𝑑 π‘˜ ⎞ ⎟ ⎟ ⎠  π‘˜ ! ∞  β„“ = 0 𝐸 ( π‘š βˆ’ 1 ) β„“ , 𝜁 𝑀 1 ξ€· 𝑀 2 𝑦 ξ€Έ 𝑀 β„“ 1 𝑑 β„“ ξƒͺ = β„“ ! ∞  𝑛 = 0 βŽ› ⎜ ⎜ ⎝ 𝑛  π‘˜ = 0 βŽ› ⎜ ⎜ ⎝ 𝑀 2 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝜁 𝑀 1 𝑖 𝐸 ( π‘š ) π‘˜ , 𝜁 𝑀 2 ξ‚΅ 𝑀 1 𝑀 π‘₯ + 1 𝑀 2 𝑖 ξ‚Ά ⎞ ⎟ ⎟ ⎠ 𝑀 π‘˜ 2 𝐸 π‘˜ ! ( π‘š βˆ’ 1 ) 𝑛 βˆ’ π‘˜ ξ€· 𝑀 2 𝑦 ξ€Έ 𝑀 1 𝑛 βˆ’ π‘˜ ⎞ ⎟ ⎟ ⎠ 𝑑 ( 𝑛 βˆ’ π‘˜ ) ! 𝑛 ! 𝑛 = 𝑛 ! ∞  𝑛 = 0 βŽ› ⎜ ⎜ ⎝ 𝑛  π‘˜ = 0 ξ‚΅ 𝑛 π‘˜ ξ‚Ά 𝑀 π‘˜ 2 𝑀 1 𝑛 βˆ’ π‘˜ 𝐸 ( π‘š βˆ’ 1 ) 𝑛 βˆ’ π‘˜ , 𝜁 𝑀 1 ξ€· 𝑀 2 𝑦 ξ€Έ 𝑀 2 βˆ’ 1  𝑖 = 0 ( βˆ’ 1 ) 𝑖 𝜁 𝑀 1 𝑖 𝐸 ( π‘š ) π‘˜ , 𝜁 𝑀 2 ξ‚΅ 𝑀 1 𝑀 π‘₯ + 1 𝑀 2 𝑖 ξ‚Ά ⎞ ⎟ ⎟ ⎠ 𝑑 𝑛 . 𝑛 ! ( 2 . 1 4 ) By comparing the coefficients on both sides of (2.13) and (2.14), we obtain the following theorem. Theorem 2.2. For  𝑀 1 , 𝑀 2 ∈ β„•   with