Table of Contents
International Journal of Combinatorics
VolumeΒ 2011Β (2011), Article IDΒ 432738, 12 pages
http://dx.doi.org/10.1155/2011/432738
Research Article

Identities of Symmetry for Generalized Euler Polynomials

Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Received 10 January 2011; Accepted 15 February 2011

Academic Editor: ChΓ­nh T.Β Hoang

Copyright Β© 2011 Dae San Kim. 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

We derive eight basic identities of symmetry in three variables related to generalized Euler polynomials and alternating generalized power sums. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the 𝑝-adic fermionic integral expression of the generating function for the generalized Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating generalized power sums.

1. Introduction and Preliminaries

Let 𝑝 be a fixed odd prime. Throughout this paper, ℀𝑝, β„šπ‘, and ℂ𝑝 will, respectively, denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of the algebraic closure of β„šπ‘. Let 𝑑 be a fixed odd positive integer. Then we let 𝑋=𝑋𝑑=limβŸ΅π‘β„€π‘‘π‘π‘β„€,(1.1) and let πœ‹βˆΆπ‘‹β†’β„€π‘ be the map given by the inverse limit of the natural mapsβ„€π‘‘π‘π‘β„€βŸΆβ„€π‘π‘β„€.(1.2) If 𝑔 is a function on ℀𝑝, then we will use the same notation to denote the function π‘”βˆ˜πœ‹. Let πœ’βˆΆ(β„€/𝑑℀)βˆ—β†’β„šβˆ— be a (primitive) Dirichlet character of conductor 𝑑. Then it will be pulled back to 𝑋 via the natural map 𝑋→℀/𝑑℀. Here we fix, once and for all, an imbedding β„šβ†’β„‚π‘, so that πœ’ is regarded as a map of 𝑋 to ℂ𝑝 (cf. [1]).

For a continuous function π‘“βˆΆπ‘‹β†’β„‚π‘, the 𝑝-adic fermionic integral of 𝑓 is defined byξ€œπ‘‹π‘“(𝑧)π‘‘πœ‡βˆ’1(𝑧)=limπ‘β†’βˆžπ‘‘π‘π‘βˆ’1𝑗=0𝑓(𝑗)(βˆ’1)𝑗.(1.3) Then it is easy to see thatξ€œπ‘‹π‘“(𝑧+1)π‘‘πœ‡βˆ’1(ξ€œπ‘§)+𝑋𝑓(𝑧)π‘‘πœ‡βˆ’1(𝑧)=2𝑓(0).(1.4) More generally, we deduce from (1.4) that, for any odd positive integer 𝑛,ξ€œπ‘‹π‘“(𝑧+𝑛)π‘‘πœ‡βˆ’1ξ€œ(𝑧)+𝑋𝑓(𝑧)π‘‘πœ‡βˆ’1(𝑧)=2π‘›βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπ‘“(π‘Ž)(1.5) and that, for any even positive integer 𝑛, ξ€œπ‘‹π‘“(𝑧+𝑛)π‘‘πœ‡βˆ’1ξ€œ(𝑧)βˆ’π‘‹π‘“(𝑧)π‘‘πœ‡βˆ’1(𝑧)=2π‘›βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žβˆ’1𝑓(π‘Ž).(1.6)

Let |β‹…|𝑝 be the normalized absolute value of ℂ𝑝, such that |𝑝|𝑝=1/𝑝, and let𝐸=π‘‘βˆˆβ„‚π‘βˆ£|𝑑|𝑝<π‘βˆ’1/(π‘βˆ’1)ξ€Ύ.(1.7) Then, for each fixed π‘‘βˆˆπΈ, the function 𝑒𝑧𝑑 is analytic on ℀𝑝 and hence considered as a function on 𝑋, and, by applying (1.5) to 𝑓 with 𝑓(𝑧)=πœ’(𝑧)𝑒𝑧𝑑, we get the 𝑝-adic integral expression of the generating function for the generalized Euler numbers 𝐸𝑛,πœ’ attached to πœ’:ξ€œπ‘‹πœ’(𝑧)π‘’π‘§π‘‘π‘‘πœ‡βˆ’12(𝑧)=𝑒𝑑𝑑+1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘‘=βˆžξ“π‘›=0𝐸𝑛,πœ’π‘‘π‘›π‘›!(π‘‘βˆˆπΈ).(1.8) So we have the following 𝑝-adic integral expression of the generating function for the generalized Euler polynomials 𝐸𝑛,πœ’(π‘₯) attached to πœ’:ξ€œπ‘‹πœ’(𝑧)𝑒(π‘₯+𝑧)π‘‘π‘‘πœ‡βˆ’1(𝑧)=2𝑒π‘₯𝑑𝑒𝑑𝑑+1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘‘=βˆžξ“π‘›=0𝐸𝑛,πœ’π‘‘(π‘₯)𝑛𝑛!π‘‘βˆˆπΈ,π‘₯βˆˆβ„€π‘ξ€Έ.(1.9) Also, from (1.4), we haveξ€œπ‘‹π‘’π‘§π‘‘π‘‘πœ‡βˆ’1(2𝑧)=𝑒𝑑(+1π‘‘βˆˆπΈ).(1.10)

Let π‘‡π‘˜(𝑛,πœ’) denote the π‘˜th alternating generalized power sum of the first 𝑛+1 nonnegative integers attached to πœ’, namely,π‘‡π‘˜(𝑛,πœ’)=π‘›ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘Žπ‘˜=(βˆ’1)0πœ’(0)0π‘˜+(βˆ’1)1πœ’(1)1π‘˜+β‹―+(βˆ’1)π‘›πœ’(𝑛)π‘›π‘˜.(1.11) From (1.8), (1.10), and (1.11), one easily derives the following identities: for any odd positive integer 𝑀,βˆ«π‘‹πœ’(π‘₯)𝑒π‘₯π‘‘π‘‘πœ‡βˆ’1(π‘₯)βˆ«π‘‹π‘’π‘€π‘‘π‘¦π‘‘π‘‘πœ‡βˆ’1=𝑒(𝑦)𝑀𝑑𝑑+1𝑒𝑑𝑑+1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘‘=(1.12)π‘€π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘‘=(1.13)βˆžξ“π‘˜=0π‘‡π‘˜π‘‘(π‘€π‘‘βˆ’1,πœ’)π‘˜π‘˜!(π‘‘βˆˆπΈ).(1.14) In what follows, we will always assume that the 𝑝-adic integrals of the various (twisted) exponential functions on 𝑋 are defined for π‘‘βˆˆπΈ (cf. (1.7)), and therefore it will not be mentioned.

References [2–6] are some of the previous works on identities of symmetry in two variables involving Bernoulli polynomials and power sums. On the other hand, for the first time we were able to produce in [7] some identities of symmetry in three variables related to Bernoulli polynomials and power sums and to extend in [8] to the case of generalized Bernoulli polynomials and generalized power sums. Also, [4] is about identities of symmetry in two variables for Euler polynomials and alternating power sums, and [9] is about those in three variables for them.

In this paper, we will be able to produce 8 identities of symmetry in three variables regarding generalized Euler polynomials and alternating generalized power sums. The case of two variables was treated in [10].

The following is stated as Theorem 4.2 and an example of the full six symmetries in 𝑀1, 𝑀2, 𝑀3:ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝐸𝑙,πœ’ξ€·π‘€2𝑦2ξ€Έπ‘‡π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’1𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€3π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝐸𝑙,πœ’ξ€·π‘€3𝑦2ξ€Έπ‘‡π‘šξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’1𝑙+π‘šπ‘€3π‘˜+π‘šπ‘€2π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€2𝑦1𝐸𝑙,πœ’ξ€·π‘€1𝑦2ξ€Έπ‘‡π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’2𝑙+π‘šπ‘€1π‘˜+π‘šπ‘€3π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€2𝑦1𝐸𝑙,πœ’ξ€·π‘€3𝑦2ξ€Έπ‘‡π‘šξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’2𝑙+π‘šπ‘€3π‘˜+π‘šπ‘€1π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€3𝑦1𝐸𝑙,πœ’ξ€·π‘€2𝑦2ξ€Έπ‘‡π‘šξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’3𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€1π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€3𝑦1𝐸𝑙,πœ’ξ€·π‘€1𝑦2ξ€Έπ‘‡π‘šξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’3𝑙+π‘šπ‘€1π‘˜+π‘šπ‘€2π‘˜+𝑙.(1.15)

The derivations of identities are based on the 𝑝-adic integral expression of the generating function for the generalized Euler polynomials in (1.9) and the quotient of integrals in (1.12) that can be expressed as the exponential generating function for the alternating generalized power sums. This abundance of symmetries would not be unearthed if such 𝑝-adic integral representations had not been available. We indebted this idea to paper [10].

2. Several Types of Quotients of 𝑝-Adic Fermionic Integrals

Here we will introduce several types of quotients of 𝑝-adic fermionic integrals on 𝑋 or 𝑋3 from which some interesting identities follow owing to the built-in symmetries in 𝑀1, 𝑀2, 𝑀3. In the following, 𝑀1, 𝑀2, 𝑀3 are all positive integers, and all of the explicit expressions of integrals in (2.2), (2.4), (2.6), and (2.8) are obtained from the identities in (1.8) and (1.10). To ease notations, from now on, we will suppress πœ‡βˆ’1 and denote, for example, π‘‘πœ‡βˆ’1(π‘₯) simply by 𝑑π‘₯.

(a) Type Λ𝑖23 (for 𝑖=0,1,2,3): 𝐼Λ𝑖23ξ€Έ=βˆ«π‘‹3πœ’ξ€·π‘₯1ξ€Έπœ’ξ€·π‘₯2ξ€Έπœ’ξ€·π‘₯3𝑒(𝑀2𝑀3π‘₯1+𝑀1𝑀3π‘₯2+𝑀1𝑀2π‘₯3+𝑀1𝑀2𝑀3(βˆ‘3βˆ’π‘–π‘—=1𝑦𝑗))𝑑𝑑π‘₯1𝑑π‘₯2𝑑π‘₯3ξ€·βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑀3π‘₯4𝑑𝑑π‘₯4𝑖=2(2.1)3βˆ’π‘–π‘’π‘€1𝑀2𝑀3(βˆ‘3βˆ’π‘–π‘—=1𝑦𝑗)𝑑𝑒𝑑𝑀1𝑀2𝑀3𝑑+1𝑖𝑒𝑑𝑀2𝑀3𝑑𝑒+1𝑑𝑀1𝑀3𝑑𝑒+1𝑑𝑀1𝑀2𝑑×+1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€2𝑀3𝑑ξƒͺξƒ©π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€1𝑀3𝑑ξƒͺξƒ©π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€1𝑀2𝑑ξƒͺ.(2.2)(b) Type Λ𝑖13 (for 𝑖=0,1,2,3): 𝐼Λ𝑖13ξ€Έ=βˆ«π‘‹3πœ’ξ€·π‘₯1ξ€Έπœ’ξ€·π‘₯2ξ€Έπœ’ξ€·π‘₯3𝑒(𝑀1π‘₯1+𝑀2π‘₯2+𝑀3π‘₯3+𝑀1𝑀2𝑀3(βˆ‘3βˆ’π‘–π‘—=1𝑦𝑗))𝑑𝑑π‘₯1𝑑π‘₯2𝑑π‘₯3ξ€·βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑀3π‘₯4𝑑𝑑π‘₯4𝑖=2(2.3)3βˆ’π‘–π‘’π‘€1𝑀2𝑀3(βˆ‘3βˆ’π‘–π‘—=1𝑦𝑗)𝑑𝑒𝑑𝑀1𝑀2𝑀3𝑑+1𝑖𝑒𝑑𝑀1𝑑𝑒+1𝑑𝑀2𝑑𝑒+1𝑑𝑀3𝑑×+1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€1𝑑ξƒͺξƒ©π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€2𝑑ξƒͺξƒ©π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€3𝑑ξƒͺ.(2.4)(c-0) Type Ξ›012: 𝐼Λ012ξ€Έ=ξ€œπ‘‹3πœ’ξ€·π‘₯1ξ€Έπœ’ξ€·π‘₯2ξ€Έπœ’ξ€·π‘₯3𝑒(𝑀1π‘₯1+𝑀2π‘₯2+𝑀3𝑀3+𝑀2𝑀3𝑦+𝑀1𝑀3𝑦+𝑀1𝑀2𝑦)𝑑𝑑π‘₯1𝑑π‘₯2𝑑π‘₯3=(2.5)8𝑒(𝑀2𝑀3+𝑀1𝑀3+𝑀1𝑀2)𝑦𝑑𝑒𝑑𝑀1𝑑𝑒+1𝑑𝑀2𝑑𝑒+1𝑑𝑀3𝑑×+1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€1𝑑ξƒͺξƒ©π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€2𝑑ξƒͺξƒ©π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€3𝑑ξƒͺ.(2.6)(c-1) Type Ξ›112: 𝐼Λ112ξ€Έ=βˆ«π‘‹3πœ’ξ€·π‘₯1ξ€Έπœ’ξ€·π‘₯2ξ€Έπœ’ξ€·π‘₯3𝑒(𝑀1π‘₯1+𝑀2π‘₯2+𝑀3π‘₯3)𝑑𝑑π‘₯1𝑑π‘₯2𝑑π‘₯3βˆ«π‘‹3𝑒𝑑(𝑀2𝑀3𝑧1+𝑀1𝑀3𝑧2+𝑀1𝑀2𝑧3)𝑑𝑑𝑧1𝑧2𝑧3=𝑒(2.7)𝑑𝑀2𝑀3𝑑𝑒+1𝑑𝑀1𝑀3𝑑𝑒+1𝑑𝑀1𝑀2𝑑+1𝑒𝑑𝑀1𝑑𝑒+1𝑑𝑀2𝑑𝑒+1𝑑𝑀3𝑑×+1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€1𝑑ξƒͺξƒ©π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€2𝑑ξƒͺξƒ©π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)π‘’π‘Žπ‘€3𝑑ξƒͺ.(2.8)

All of the above 𝑝-adic integrals of various types are invariant under all permutations of 𝑀1, 𝑀2, 𝑀3, as one can see either from 𝑝-adic integral representations in (2.1), (2.3), (2.5), and (2.7) or from their explicit evaluations in (2.2), (2.4), (2.6), and (2.8).

3. Identities for Generalized Euler Polynomials

In the following, 𝑀1, 𝑀2, 𝑀3 are all odd positive integers except for (a-0) and (c-0), where they are any positive integers. First, let's consider Type Λ𝑖23, for each 𝑖=0,1,2,3. The following results can be easily obtained from (1.9) and (1.12):

(a-0)𝐼Λ023ξ€Έ=ξ€œπ‘‹πœ’ξ€·π‘₯1𝑒𝑀2𝑀3(π‘₯1+𝑀1𝑦1)𝑑𝑑π‘₯1ξ€œπ‘‹πœ’ξ€·π‘₯2𝑒𝑀1𝑀3(π‘₯2+𝑀2𝑦2)𝑑𝑑π‘₯2ξ€œπ‘‹πœ’ξ€·π‘₯3𝑒𝑀1𝑀2(π‘₯3+𝑀3𝑦3)𝑑𝑑π‘₯3=ξƒ©βˆžξ“π‘˜=0πΈπ‘˜,πœ’ξ€·π‘€1𝑦1ξ€Έξ€·π‘€π‘˜!2𝑀3π‘‘ξ€Έπ‘˜ξƒͺξƒ©βˆžξ“π‘™=0𝐸𝑙,πœ’ξ€·π‘€2𝑦2𝑀𝑙!1𝑀3𝑑𝑙ξƒͺξƒ©βˆžξ“π‘š=0πΈπ‘š,πœ’ξ€·π‘€3𝑦3ξ€Έξ€·π‘€π‘š!1𝑀2π‘‘ξ€Έπ‘šξƒͺ=βˆžξ“π‘›=0ξƒ©ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝐸𝑙,πœ’ξ€·π‘€2𝑦2ξ€ΈπΈπ‘š,πœ’ξ€·π‘€3𝑦3𝑀1𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€3π‘˜+𝑙ξƒͺ𝑑𝑛,𝑛!(3.1) where the inner sum is over all nonnegative integers π‘˜, 𝑙, π‘š with π‘˜+𝑙+π‘š=𝑛 and 𝑛=π‘˜,𝑙,π‘šπ‘›!π‘˜!𝑙!π‘š!.(3.2)(a-1) Here we write 𝐼(Ξ›123) in two different ways: (1)𝐼Λ123ξ€Έ=ξ€œπ‘‹πœ’ξ€·π‘₯1𝑒𝑀2𝑀3(π‘₯1+𝑀1𝑦1)𝑑𝑑π‘₯1ξ€œπ‘‹πœ’ξ€·π‘₯2𝑒𝑀1𝑀3(π‘₯2+𝑀2𝑦2)𝑑𝑑π‘₯2βˆ«π‘‹πœ’ξ€·π‘₯3𝑒𝑀1𝑀2π‘₯3𝑑𝑑π‘₯3βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑀3π‘₯4𝑑𝑑π‘₯4=ξƒ©βˆžξ“π‘˜=0πΈπ‘˜,πœ’ξ€·π‘€1𝑦1𝑀2𝑀3π‘‘ξ€Έπ‘˜π‘˜!ξƒͺξƒ©βˆžξ“π‘™=0𝐸𝑙,πœ’ξ€·π‘€2𝑦2𝑀1𝑀3𝑑𝑙ξƒͺ×𝑇𝑙!π‘šξ€·π‘€3ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’1𝑀2π‘‘ξ€Έπ‘šξƒͺ=π‘š!(3.3)βˆžξ“π‘›=0ξƒ©ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝐸𝑙,πœ’ξ€·π‘€2𝑦2ξ€Έπ‘‡π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’1𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€3π‘˜+𝑙ξƒͺ𝑑𝑛.𝑛!(3.4)(2) Invoking (1.13), (3.3) can also be written as 𝐼Λ123ξ€Έ=𝑀3π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žξ€œπœ’(π‘Ž)π‘‹πœ’ξ€·π‘₯1𝑒𝑀2𝑀3(π‘₯1+𝑀1𝑦1)𝑑𝑑π‘₯1ξ€œπ‘‹πœ’ξ€·π‘₯2𝑒𝑀1𝑀3(π‘₯2+𝑀2𝑦2+𝑀2/𝑀3π‘Ž)𝑑𝑑π‘₯2=𝑀3π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žξƒ©πœ’(π‘Ž)βˆžξ“π‘˜=0πΈπ‘˜,πœ’ξ€·π‘€1𝑦1𝑀2𝑦3π‘‘ξ€Έπ‘˜ξƒͺΓ—ξƒ©π‘˜!βˆžξ“π‘™=0𝐸𝑙,πœ’ξ‚΅π‘€2𝑦2+𝑀2𝑀3π‘Žξ‚Άξ€·π‘€1𝑦3𝑑𝑙ξƒͺ=𝑙!βˆžξ“π‘›=0βŽ›βŽœβŽœβŽπ‘€π‘›3π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚ΆπΈπ‘˜,πœ’ξ€·π‘€1𝑦1𝑀3π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘›βˆ’π‘˜,πœ’ξ‚΅π‘€2𝑦2+𝑀2𝑀3π‘Žξ‚Άπ‘€1π‘›βˆ’π‘˜π‘€π‘˜2βŽžβŽŸβŽŸβŽ π‘‘π‘›.𝑛!(3.5)(a-2) Here we write 𝐼(Ξ›223) in three different ways: (1)𝐼Λ223ξ€Έ=ξ€œπ‘‹πœ’ξ€·π‘₯1𝑒𝑀2𝑀3(π‘₯1+𝑀1𝑦1)𝑑𝑑π‘₯1βˆ«π‘‹πœ’ξ€·π‘₯2𝑒𝑀1𝑀3π‘₯2𝑑𝑑π‘₯2βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑀3π‘₯4𝑑𝑑π‘₯4βˆ«π‘‹πœ’ξ€·π‘₯3𝑒𝑀1𝑀2π‘₯3𝑑𝑑π‘₯3βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑀3π‘₯4𝑑𝑑π‘₯4=ξƒ©βˆžξ“π‘˜=0πΈπ‘˜,πœ’ξ€·π‘€1𝑦1𝑀2𝑀3π‘‘ξ€Έπ‘˜π‘˜!ξƒͺξƒ©βˆžξ“π‘™=0𝑇𝑙𝑀2ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’1𝑀3𝑑𝑙ξƒͺ×𝑙!βˆžξ“π‘š=0π‘‡π‘šξ€·π‘€3ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’1𝑀2π‘‘ξ€Έπ‘šξƒͺ=π‘š!(3.6)βˆžξ“π‘›=0ξƒ©ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝑇𝑙𝑀2ξ€ΈΓ—ξ€·π‘€π‘‘βˆ’1,πœ’3ξ€Έπ‘€π‘‘βˆ’1,πœ’1𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€3π‘˜+𝑙ξƒͺπ‘‡π‘šπ‘‘π‘›.𝑛!(3.7)(2) Invoking (1.13), (3.6) can also be written as 𝐼Λ223ξ€Έ=𝑀2π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žξ€œπœ’(π‘Ž)π‘‹πœ’ξ€·π‘₯1𝑒𝑀2𝑀3(π‘₯1+𝑀1𝑦1+(𝑀1/𝑀2)π‘Ž)𝑑𝑑π‘₯1Γ—βˆ«π‘‹πœ’ξ€·π‘₯3𝑒𝑀1𝑀2π‘₯3𝑑𝑑π‘₯3βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑀3π‘₯4𝑑𝑑π‘₯4=𝑀2π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žξƒ©πœ’(π‘Ž)βˆžξ“π‘˜=0πΈπ‘˜,πœ’ξ‚΅π‘€1𝑦1+𝑀1𝑀2π‘Žξ‚Άξ€·π‘€2𝑀3π‘‘ξ€Έπ‘˜ξƒͺΓ—ξƒ©π‘˜!βˆžξ“π‘™=0𝑇𝑙𝑀3ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’1𝑀2𝑑𝑙ξƒͺ=𝑙!(3.8)βˆžξ“π‘›=0βŽ›βŽœβŽœβŽπ‘€π‘›2π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚Άπ‘€2π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘˜,πœ’ξ‚΅π‘€1𝑦1+𝑀1𝑀2π‘Žξ‚Άπ‘‡π‘›βˆ’π‘˜Γ—ξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’1π‘›βˆ’π‘˜π‘€π‘˜3βŽžβŽŸβŽŸβŽ π‘‘π‘›.𝑛!(3.9)(3) Invoking (1.13) once again, (3.8) can be written as 𝐼Λ223ξ€Έ=𝑀2π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)𝑀3π‘‘βˆ’1𝑏=0(βˆ’1)π‘ξ€œπœ’(𝑏)π‘‹πœ’ξ€·π‘₯1𝑒𝑀2𝑀3(π‘₯1+𝑀1𝑦1+(𝑀1/𝑀2π‘Ž)+(𝑀1/𝑀3)𝑏)𝑑𝑑π‘₯1=𝑀2π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)𝑀3π‘‘βˆ’1𝑏=0(βˆ’1)π‘πœ’(𝑏)βˆžξ“π‘›=0𝐸𝑛,πœ’ξ‚΅π‘€1𝑦1+𝑀1𝑀2π‘€π‘Ž+1𝑀3𝑏𝑀2𝑀3𝑑𝑛=𝑛!(3.10)βˆžξ“π‘›=0βŽ›βŽœβŽœβŽξ€·π‘€2𝑀3𝑛𝑀2π‘‘βˆ’1ξ“π‘€π‘Ž=03π‘‘βˆ’1𝑏=0(βˆ’1)π‘Ž+π‘πœ’(π‘Žπ‘)𝐸𝑛,πœ’ξ‚΅π‘€1𝑦1+𝑀1𝑀2π‘€π‘Ž+1𝑀3π‘ξ‚ΆβŽžβŽŸβŽŸβŽ π‘‘π‘›π‘›!.(3.11)(a-3)𝐼Λ323ξ€Έ=βˆ«π‘‹πœ’ξ€·π‘₯1𝑒𝑀2𝑀3π‘₯1𝑑𝑑π‘₯1βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑀3π‘₯4𝑑𝑑π‘₯4Γ—βˆ«π‘‹πœ’ξ€·π‘₯2𝑒𝑀1𝑀3π‘₯2𝑑𝑑π‘₯2βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑀3π‘₯4𝑑𝑑π‘₯4Γ—βˆ«π‘‹πœ’ξ€·π‘₯3𝑒𝑀1𝑀2π‘₯3𝑑𝑑π‘₯3βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑀3π‘₯4𝑑𝑑π‘₯4=ξƒ©βˆžξ“π‘˜=0π‘‡π‘˜ξ€·π‘€1ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’2𝑀3π‘‘ξ€Έπ‘˜π‘˜!ξƒͺξƒ©βˆžξ“π‘™=0𝑇𝑙𝑀2ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’1𝑀3𝑑𝑙ξƒͺ×𝑙!βˆžξ“π‘š=0π‘‡π‘šξ€·π‘€3ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’1𝑀2π‘‘ξ€Έπ‘šξƒͺ=π‘š!(3.12)βˆžξ“π‘›=0ξƒ©ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚Άπ‘‡π‘˜,𝑙,π‘šπ‘˜ξ€·π‘€1ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘™ξ€·π‘€2ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’1𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€3π‘˜+𝑙ξƒͺ𝑑𝑛.𝑛!(3.13)(b) For Type Λ𝑖13 (𝑖=0,1,2,3), we may consider the analogous things to the ones in (a-0), (a-1), (a-2), and (a-3). However, these do not lead us to new identities. Indeed, if we substitute 𝑀2𝑀3, 𝑀1𝑀3, 𝑀1𝑀2, respectively, for 𝑀1, 𝑀2, 𝑀3 in (2.1), this amounts to replacing 𝑑 by 𝑀1𝑀2𝑀3𝑑 in (2.3). So, upon replacing 𝑀1, 𝑀2, 𝑀3, respectively, by 𝑀2𝑀3, 𝑀1𝑀3, 𝑀1𝑀2 and dividing by (𝑀1𝑀2𝑀3)𝑛, in each of the expressions of (3.1), (3.4), (3.5), (3.7), (3.9)–(3.13), we will get the corresponding symmetric identities for Type Λ𝑖13 (𝑖=0,1,2,3).(c-0)𝐼Λ012ξ€Έ=ξ€œπ‘‹πœ’ξ€·π‘₯1𝑒𝑀1(π‘₯1+𝑀2𝑦)𝑑𝑑π‘₯1ξ€œπ‘‹πœ’ξ€·π‘₯2𝑒𝑀2(π‘₯2+𝑀3𝑦)𝑑𝑑π‘₯2ξ€œπ‘‹πœ’ξ€·π‘₯3𝑒𝑀3(π‘₯3+𝑀1𝑦)𝑑𝑑π‘₯3=ξƒ©βˆžξ“π‘˜=0πΈπ‘˜,πœ’ξ€·π‘€2π‘¦ξ€Έξ€·π‘€π‘˜!1π‘‘ξ€Έπ‘˜ξƒͺξƒ©βˆžξ“π‘™=0𝐸𝑙,πœ’ξ€·π‘€3𝑦𝑀𝑙!2𝑑𝑙ξƒͺξƒ©βˆžξ“π‘š=0πΈπ‘š,πœ’ξ€·π‘€1π‘¦ξ€Έξ€·π‘€π‘š!3π‘‘ξ€Έπ‘šξƒͺ=βˆžξ“π‘›=0ξƒ©ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€2𝑦𝐸𝑙,πœ’ξ€·π‘€3π‘¦ξ€ΈπΈπ‘š,πœ’ξ€·π‘€1π‘¦ξ€Έπ‘€π‘˜1𝑀𝑙2π‘€π‘š3ξƒͺ𝑑𝑛.𝑛!(3.14)(c-1)𝐼Λ112ξ€Έ=βˆ«π‘‹πœ’ξ€·π‘₯1𝑒𝑀1π‘₯1𝑑𝑑π‘₯1βˆ«π‘‹π‘’π‘‘π‘€1𝑀2𝑧3𝑑𝑑𝑧3Γ—βˆ«π‘‹πœ’ξ€·π‘₯2𝑒𝑀2π‘₯2𝑑𝑑π‘₯2βˆ«π‘‹π‘’π‘‘π‘€2𝑀3𝑧1𝑑𝑑𝑧1Γ—βˆ«π‘‹πœ’ξ€·π‘₯3𝑒𝑀3π‘₯3𝑑𝑑π‘₯3βˆ«π‘‹π‘’π‘‘π‘€3𝑀1𝑧2𝑑𝑑𝑧2=ξƒ©βˆžξ“π‘˜=0π‘‡π‘˜ξ€·π‘€2ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’1π‘‘ξ€Έπ‘˜π‘˜!ξƒͺξƒ©βˆžξ“π‘™=0𝑇𝑙𝑀3ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’2𝑑𝑙𝑙!ξƒͺξƒ©βˆžξ“π‘š=0π‘‡π‘šξ€·π‘€1ξ€Έξ€·π‘€π‘‘βˆ’1,πœ’3π‘‘ξ€Έπ‘šξƒͺ=π‘š!βˆžξ“π‘›=0ξƒ©ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚Άπ‘‡π‘˜,𝑙,π‘šπ‘˜ξ€·π‘€2ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘™ξ€·π‘€3ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’π‘˜1𝑀𝑙2π‘€π‘š3ξƒͺ𝑑𝑛.𝑛!(3.15)

4. Main Theorems

As we noted earlier in the last paragraph of Section 2, the various types of quotients of 𝑝-adic fermionic integrals are invariant under any permutation of 𝑀1, 𝑀2, 𝑀3. So the corresponding expressions in Section 3 are also invariant under any permutation of 𝑀1, 𝑀2, 𝑀3. Thus, our results about identities of symmetry will be immediate consequences of this observation.

However, not all permutations of an expression in Section 3 yield distinct ones. In fact, as these expressions are obtained by permuting 𝑀1, 𝑀2, 𝑀3 in a single one labeled by them, they can be viewed as a group in a natural manner, and hence it is isomorphic to a quotient of 𝑆3. In particular, the number of possible distinct expressions is 1, 2, 3, or 6 (a-0), (a-1(1)), (a-1(2)), and (a-2(2)) give the full six identities of symmetry, (a-2(1)) and (a-2(3)) yield three identities of symmetry, and (c-0) and (c-1) give two identities of symmetry, while the expression in (a-3) yields no identities of symmetry.

Here we will just consider the cases of Theorems 4.4 and 4.8, leaving the others as easy exercises for the reader. As for the case of Theorem 4.4, in addition to (4.11)–(4.13), we get the following three ones:ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝑇𝑙𝑀3ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’1𝑙+π‘šπ‘€3π‘˜+π‘šπ‘€2π‘˜+𝑙=(4.1)π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€2𝑦1𝑇𝑙𝑀1ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’2𝑙+π‘šπ‘€1π‘˜+π‘šπ‘€3π‘˜+𝑙=(4.2)π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€3𝑦1𝑇𝑙𝑀2ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’3𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€1π‘˜+𝑙.(4.3) But, by interchanging 𝑙 and π‘š, we see that (4.1), (4.2), and (4.3) are, respectively, equal to (4.11), (4.12), and (4.13). As to Theorem 4.8, in addition to (4.17) and (4.18), we have:ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚Άπ‘‡π‘˜,𝑙,π‘šπ‘˜ξ€·π‘€2ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘™ξ€·π‘€3ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’π‘˜1𝑀𝑙2π‘€π‘š3=(4.4)π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚Άπ‘‡π‘˜,𝑙,π‘šπ‘˜ξ€·π‘€3ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘™ξ€·π‘€1ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’π‘˜2𝑀𝑙3π‘€π‘š1=(4.5)π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚Άπ‘‡π‘˜,𝑙,π‘šπ‘˜ξ€·π‘€3ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘™ξ€·π‘€2ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’π‘˜1𝑀𝑙3π‘€π‘š2=(4.6)π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚Άπ‘‡π‘˜,𝑙,π‘šπ‘˜ξ€·π‘€2ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘™ξ€·π‘€1ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’π‘˜3𝑀𝑙2π‘€π‘š1.(4.7) However, (4.4) and (4.5) are equal to (4.17), as we can see by applying the permutations π‘˜β†’π‘™, π‘™β†’π‘š, π‘šβ†’π‘˜ for (4.4) and π‘˜β†’π‘š, π‘™β†’π‘˜, π‘šβ†’π‘™ for (4.5). Similarly, we see that (4.6) and (4.7) are equal to (4.18), by applying permutations π‘˜β†’π‘™, π‘™β†’π‘š, π‘šβ†’π‘˜ for (4.6) and π‘˜β†’π‘š, π‘™β†’π‘˜, π‘šβ†’π‘™ for (4.7).

Theorem 4.1. Let 𝑀1, 𝑀2, 𝑀3 be any positive integers. Then one has ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝐸𝑙,πœ’ξ€·π‘€2𝑦2ξ€ΈπΈπ‘š,πœ’ξ€·π‘€3𝑦3𝑀1𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€3π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝐸𝑙,πœ’ξ€·π‘€3𝑦2ξ€ΈπΈπ‘š,πœ’ξ€·π‘€2𝑦3𝑀1𝑙+π‘šπ‘€3π‘˜+π‘šπ‘€2π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€2𝑦1𝐸𝑙,πœ’ξ€·π‘€1𝑦2ξ€ΈπΈπ‘š,πœ’ξ€·π‘€3𝑦3𝑀2𝑙+π‘šπ‘€1π‘˜+π‘šπ‘€3π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€2𝑦1𝐸𝑙,πœ’ξ€·π‘€3𝑦2ξ€ΈπΈπ‘š,πœ’ξ€·π‘€1𝑦3𝑀2𝑙+π‘šπ‘€3π‘˜+π‘šπ‘€1π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€3𝑦1𝐸𝑙,πœ’ξ€·π‘€1𝑦2ξ€ΈπΈπ‘š,πœ’ξ€·π‘€2𝑦3𝑀3𝑙+π‘šπ‘€1π‘˜+π‘šπ‘€2π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,mπ‘˜,πœ’ξ€·π‘€3𝑦1𝐸𝑙,πœ’ξ€·π‘€2𝑦2ξ€ΈπΈπ‘š,πœ’ξ€·π‘€1𝑦3𝑀3𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€1π‘˜+𝑙.(4.8)

Theorem 4.2. Let 𝑀1, 𝑀2, 𝑀3 be any odd positive integers. Then one has ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝐸𝑙,πœ’ξ€·π‘€2𝑦2ξ€Έπ‘‡π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’1𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€3π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝐸𝑙,πœ’ξ€·π‘€3𝑦2ξ€Έπ‘‡π‘šξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’1𝑙+π‘šπ‘€3π‘˜+π‘šπ‘€2π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€2𝑦1𝐸𝑙,πœ’ξ€·π‘€1𝑦2ξ€Έπ‘‡π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’2𝑙+π‘šπ‘€1π‘˜+π‘šπ‘€3π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€2𝑦1𝐸𝑙,πœ’ξ€·π‘€3𝑦2ξ€Έπ‘‡π‘šξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’2𝑙+π‘šπ‘€3π‘˜+π‘šπ‘€1π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€3𝑦1𝐸𝑙,πœ’ξ€·π‘€2𝑦2ξ€Έπ‘‡π‘šξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’3𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€1π‘˜+𝑙=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€3𝑦1𝐸𝑙,πœ’ξ€·π‘€1𝑦2ξ€Έπ‘‡π‘šξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’3𝑙+π‘šπ‘€1π‘˜+π‘šπ‘€2π‘˜+𝑙.(4.9)

Theorem 4.3. Let 𝑀1, 𝑀2, 𝑀3 be any odd positive integers. Then one has 𝑀𝑛1π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚ΆπΈπ‘˜,πœ’ξ€·π‘€3𝑦1𝑀1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘›βˆ’π‘˜,πœ’ξ‚΅π‘€2𝑦2+𝑀2𝑀1π‘Žξ‚Άπ‘€3π‘›βˆ’π‘˜π‘€π‘˜2=𝑀𝑛1π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚ΆπΈπ‘˜,πœ’ξ€·π‘€2𝑦1𝑀1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘›βˆ’π‘˜,πœ’ξ‚΅π‘€3𝑦2+𝑀3𝑀1π‘Žξ‚Άπ‘€2π‘›βˆ’π‘˜π‘€π‘˜3=𝑀𝑛2π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚ΆπΈπ‘˜,πœ’ξ€·π‘€3𝑦1𝑀2π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘›βˆ’π‘˜,πœ’ξ‚΅π‘€1𝑦2+𝑀1𝑀2π‘Žξ‚Άπ‘€3π‘›βˆ’π‘˜π‘€π‘˜1=𝑀𝑛2π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚ΆπΈπ‘˜,πœ’ξ€·π‘€1𝑦1𝑀2π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘›βˆ’π‘˜,πœ’ξ‚΅π‘€3𝑦2+𝑀3𝑀2π‘Žξ‚Άπ‘€1π‘›βˆ’π‘˜π‘€π‘˜3=𝑀𝑛3π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚ΆπΈπ‘˜,πœ’ξ€·π‘€2𝑦1𝑀3π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘›βˆ’π‘˜,πœ’ξ‚΅π‘€1𝑦2+𝑀1𝑀3π‘Žξ‚Άπ‘€2π‘›βˆ’π‘˜π‘€π‘˜1=𝑀𝑛3π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚ΆπΈπ‘˜,πœ’ξ€·π‘€1𝑦1𝑀3π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘›βˆ’π‘˜,πœ’ξ‚΅π‘€2𝑦2+𝑀2𝑀3π‘Žξ‚Άπ‘€1π‘›βˆ’π‘˜π‘€π‘˜2.(4.10)

Theorem 4.4. Let 𝑀1, 𝑀2, 𝑀3 be any odd positive integers. Then one has the following three symmetries in 𝑀1, 𝑀2, 𝑀3: ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦1𝑇𝑙𝑀2ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’1𝑙+π‘šπ‘€2π‘˜+π‘šπ‘€3π‘˜+𝑙=(4.11)π‘˜+𝑙+π‘š=𝑛𝑛Eπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€2𝑦1𝑇𝑙𝑀3ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’2𝑙+π‘šπ‘€3π‘˜+π‘šπ‘€1π‘˜+𝑙=(4.12)π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€3𝑦1𝑇𝑙𝑀1ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’3𝑙+π‘šπ‘€1π‘˜+π‘šπ‘€2π‘˜+𝑙.(4.13)

Theorem 4.5. Let 𝑀1, 𝑀2, 𝑀3 be any odd positive integers. Then one has 𝑀𝑛1π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚Άπ‘€1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘˜,πœ’ξ‚΅π‘€2𝑦1+𝑀2𝑀1π‘Žξ‚Άπ‘‡π‘›βˆ’π‘˜ξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’2π‘›βˆ’π‘˜π‘€π‘˜3=𝑀𝑛1π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚Άπ‘€1π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘˜,πœ’ξ‚΅π‘€3𝑦1+𝑀3𝑀1π‘Žξ‚Άπ‘‡π‘›βˆ’π‘˜ξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’3π‘›βˆ’π‘˜π‘€π‘˜2=𝑀𝑛2π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚Άπ‘€2π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘˜,πœ’ξ‚΅π‘€1𝑦1+𝑀1𝑀2π‘Žξ‚Άπ‘‡π‘›βˆ’π‘˜ξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’1π‘›βˆ’π‘˜π‘€π‘˜3=𝑀𝑛2π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚Άπ‘€2π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘˜,πœ’ξ‚΅π‘€3𝑦1+𝑀3𝑀2π‘Žξ‚Άπ‘‡π‘›βˆ’π‘˜ξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’3π‘›βˆ’π‘˜π‘€π‘˜1=𝑀𝑛3π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚Άπ‘€3π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘˜,πœ’ξ‚΅π‘€1𝑦1+𝑀1𝑀3π‘Žξ‚Άπ‘‡π‘›βˆ’π‘˜ξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’1π‘›βˆ’π‘˜π‘€π‘˜2=𝑀𝑛3π‘›ξ“π‘˜=0ξ‚΅π‘›π‘˜ξ‚Άπ‘€3π‘‘βˆ’1ξ“π‘Ž=0(βˆ’1)π‘Žπœ’(π‘Ž)πΈπ‘˜,πœ’ξ‚΅π‘€2𝑦1+𝑀2𝑀3π‘Žξ‚Άπ‘‡π‘›βˆ’π‘˜ξ€·π‘€1ξ€Έπ‘€π‘‘βˆ’1,πœ’2π‘›βˆ’π‘˜π‘€k1.(4.14)

Theorem 4.6. Let 𝑀1, 𝑀2, 𝑀3 be any odd positive integers. Then one has the following three symmetries in 𝑀1, 𝑀2, 𝑀3: 𝑀1𝑀2𝑛𝑀1π‘‘βˆ’1ξ“π‘€π‘Ž=02π‘‘βˆ’1𝑏=0(βˆ’1)π‘Ž+π‘πœ’(π‘Žπ‘)𝐸𝑛,πœ’ξ‚΅π‘€3𝑦1+𝑀3𝑀1π‘€π‘Ž+3𝑀2𝑏=𝑀2𝑀3𝑛𝑀2π‘‘βˆ’1ξ“π‘€π‘Ž=03π‘‘βˆ’1𝑏=0(βˆ’1)π‘Ž+π‘πœ’(π‘Žπ‘)𝐸𝑛,πœ’ξ‚΅π‘€1𝑦1+𝑀1𝑀2π‘€π‘Ž+1𝑀3𝑏=𝑀3𝑀1𝑛𝑀3π‘‘βˆ’1ξ“π‘€π‘Ž=01π‘‘βˆ’1𝑏=0(βˆ’1)π‘Ž+π‘πœ’(π‘Žπ‘)𝐸𝑛,πœ’ξ‚΅π‘€2𝑦1+𝑀2𝑀3π‘€π‘Ž+2𝑀1𝑏.(4.15)

Theorem 4.7. Let 𝑀1, 𝑀2, 𝑀3 be any positive integers. Then one has the following two symmetries in 𝑀1, 𝑀2, 𝑀3: ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦𝐸𝑙,πœ’ξ€·π‘€2π‘¦ξ€ΈπΈπ‘š,πœ’ξ€·π‘€3π‘¦ξ€Έπ‘€π‘˜3𝑀𝑙1π‘€π‘š2=ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚ΆπΈπ‘˜,𝑙,π‘šπ‘˜,πœ’ξ€·π‘€1𝑦𝐸𝑙,πœ’ξ€·π‘€3π‘¦ξ€ΈπΈπ‘š,πœ’ξ€·π‘€2π‘¦ξ€Έπ‘€π‘˜2𝑀𝑙1π‘€π‘š3.(4.16)

Theorem 4.8. Let 𝑀1, 𝑀2, 𝑀3 be any odd positive integers. Then one has the following two symmetries in 𝑀1, 𝑀2, 𝑀3: ξ“π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚Άπ‘‡π‘˜,𝑙,π‘šπ‘˜ξ€·π‘€1ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘™ξ€·π‘€2ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€3ξ€Έπ‘€π‘‘βˆ’1,πœ’π‘˜3𝑀𝑙1π‘€π‘š2,(4.17)π‘˜+𝑙+π‘š=π‘›ξ‚΅π‘›ξ‚Άπ‘‡π‘˜,𝑙,π‘šπ‘˜ξ€·π‘€1ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘™ξ€·π‘€3ξ€Έπ‘‡π‘‘βˆ’1,πœ’π‘šξ€·π‘€2ξ€Έπ‘€π‘‘βˆ’1,πœ’π‘˜2𝑀𝑙1π‘€π‘š3.(4.18)

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