Identities of Symmetry for Generalized Euler Polynomials

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, ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will, respectively, denote the ring of 𝑝-adic integers, the field of 𝑝-adic rational numbers, and the completion of the algebraic closure of ℚ_𝑝. Let 𝑑 be a fixed odd positive integer. Then we let 𝑋=𝑋_𝑑=lim_⟵𝑁ℤ𝑑𝑝^𝑁ℤ,(1.1) and let 𝜋∶𝑋→ℤ_𝑝 be the map given by the inverse limit of the natural mapsℤ𝑑𝑝^𝑁ℤ⟶𝑝^𝑁ℤℤ.(1.2) If 𝑔 is a function on ℤ_𝑝, then we will use the same notation to denote the function 𝑔∘𝜋. Let 𝜒∶(ℤ/𝑑ℤ)^∗→ℚ^∗ be a (primitive) Dirichlet character of conductor 𝑑. Then it will be pulled back to 𝑋 via the natural map 𝑋→ℤ/𝑑ℤ. Here we fix, once and for all, an imbedding ℚ→ℂ_𝑝, so that 𝜒 is regarded as a map of 𝑋 to ℂ_𝑝 (cf. [1]).

For a continuous function 𝑓∶𝑋→ℂ_𝑝, the 𝑝-adic fermionic integral of 𝑓 is defined by_𝑋𝑓(𝑧)𝑑𝜇₋₁(𝑧)=lim_{𝑁→∞𝑑𝑝}^𝑁−1_𝑗=0𝑓(𝑗)(−1)^𝑗.(1.3) Then it is easy to see that_𝑋𝑓(𝑧+1)𝑑𝜇₋₁(𝑧)+_𝑋𝑓(𝑧)𝑑𝜇₋₁(𝑧)=2𝑓(0).(1.4) More generally, we deduce from (1.4) that, for any odd positive integer 𝑛,_𝑋𝑓(𝑧+𝑛)𝑑𝜇₋₁(𝑧)+_𝑋𝑓(𝑧)𝑑𝜇₋₁(𝑧)=2^𝑛−1_𝑎=0(−1)^𝑎𝑓(𝑎)(1.5) and that, for any even positive integer 𝑛, _𝑋𝑓(𝑧+𝑛)𝑑𝜇₋₁(𝑧)−_𝑋𝑓(𝑧)𝑑𝜇₋₁(𝑧)=2^𝑛−1_𝑎=0(−1)^𝑎−1𝑓(𝑎).(1.6)

Let |⋅|_𝑝 be the normalized absolute value of ℂ_𝑝, such that |𝑝|_𝑝=1/𝑝, and let𝐸=𝑡∈ℂ_𝑝∣|𝑡|_𝑝<𝑝^{−1/(𝑝−1)}.(1.7) Then, for each fixed 𝑡∈𝐸, the function 𝑒^𝑧𝑡 is analytic on ℤ_𝑝 and hence considered as a function on 𝑋, and, by applying (1.5) to 𝑓 with 𝑓(𝑧)=𝜒(𝑧)𝑒^𝑧𝑡, we get the 𝑝-adic integral expression of the generating function for the generalized Euler numbers 𝐸_𝑛,𝜒 attached to 𝜒:_𝑋𝜒(𝑧)𝑒^𝑧𝑡𝑑𝜇₋₁(𝑧)=2𝑒^𝑑𝑡+1^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^𝑎𝑡=^∞_𝑛=0𝐸_𝑛,𝜒𝑡^𝑛𝑛!(𝑡∈𝐸).(1.8) So we have the following 𝑝-adic integral expression of the generating function for the generalized Euler polynomials 𝐸_𝑛,𝜒(𝑥) attached to 𝜒:_𝑋𝜒(𝑧)𝑒^{(𝑥+𝑧)𝑡}𝑑𝜇₋₁(𝑧)=2𝑒^𝑥𝑡𝑒^𝑑𝑡+1^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^𝑎𝑡=^∞_𝑛=0𝐸_𝑛,𝜒(𝑥)𝑡^𝑛𝑛!𝑡∈𝐸,𝑥∈ℤ^𝑝.(1.9) Also, from (1.4), we have_𝑋𝑒^𝑧𝑡𝑑𝜇₋₁(𝑧)=2𝑒^𝑡+1(𝑡∈𝐸).(1.10)

Let 𝑇_𝑘(𝑛,𝜒) denote the 𝑘th alternating generalized power sum of the first 𝑛+1 nonnegative integers attached to 𝜒, namely,𝑇_𝑘(𝑛,𝜒)=^𝑛_𝑎=0(−1)^𝑎𝜒(𝑎)𝑎^𝑘=(−1)⁰𝜒(0)0^𝑘+(−1)¹𝜒(1)1^𝑘+⋯+(−1)^𝑛𝜒(𝑛)𝑛^𝑘.(1.11) From (1.8), (1.10), and (1.11), one easily derives the following identities: for any odd positive integer 𝑤,∫_𝑋𝜒(𝑥)𝑒^𝑥𝑡𝑑𝜇₋₁(𝑥)∫_𝑋𝑒^{𝑤𝑑𝑦𝑡}𝑑𝜇₋₁(𝑦)=𝑒^𝑤𝑑𝑡+1𝑒^𝑑𝑡+1^𝑑−1_𝑎=0=(−1)^𝑎𝜒(𝑎)𝑒^𝑎𝑡(1.12)_𝑤𝑑−1_𝑎=0=(−1)^𝑎𝜒(𝑎)𝑒^𝑎𝑡(1.13)_∞_𝑘=0𝑇_𝑘(𝑤𝑑−1,𝜒)𝑡^𝑘𝑘!(𝑡∈𝐸).(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 𝑡∈𝐸 (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 𝑤₁, 𝑤₂, 𝑤₃:_𝑘+_{𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦₁𝐸_𝑙,𝜒𝑤₂𝑦₂𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^1𝑙+𝑚𝑤^2𝑘+𝑚𝑤^3𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦₁𝐸_𝑙,𝜒𝑤₃𝑦₂𝑇_𝑚𝑤₂𝑑−1,𝜒𝑤^1𝑙+𝑚𝑤^3𝑘+𝑚𝑤^2𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₂𝑦₁𝐸_𝑙,𝜒𝑤₁𝑦₂𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^2𝑙+𝑚𝑤^1𝑘+𝑚𝑤^3𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₂𝑦₁𝐸_𝑙,𝜒𝑤₃𝑦₂𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤^2𝑙+𝑚𝑤^3𝑘+𝑚𝑤^1𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₃𝑦₁𝐸_𝑙,𝜒𝑤₂𝑦₂𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤^3𝑙+𝑚𝑤^2𝑘+𝑚𝑤^1𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₃𝑦₁𝐸_𝑙,𝜒𝑤₁𝑦₂𝑇_𝑚𝑤₂𝑑−1,𝜒𝑤^3𝑙+𝑚𝑤^1𝑘+𝑚𝑤^2𝑘+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 𝑤₁, 𝑤₂, 𝑤₃. In the following, 𝑤₁, 𝑤₂, 𝑤₃ 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 𝜇₋₁ and denote, for example, 𝑑𝜇₋₁(𝑥) simply by 𝑑𝑥.

(a) Type Λ^𝑖23 (for 𝑖=0,1,2,3): 𝐼Λ^𝑖23=∫^𝑋3𝜒𝑥₁𝜒𝑥₂𝜒𝑥₃𝑒^{(𝑤2𝑤3𝑥1+𝑤1𝑤3𝑥2+𝑤1𝑤2𝑥3+𝑤1𝑤2𝑤3(∑3−𝑖𝑗=1𝑦𝑗))𝑡}𝑑𝑥₁𝑑𝑥₂𝑑𝑥₃∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑤3𝑥4𝑡}𝑑𝑥₄^𝑖=2(2.1)_3−𝑖𝑒^{𝑤1𝑤2𝑤3(∑3−𝑖𝑗=1𝑦𝑗)𝑡}𝑒^{𝑑𝑤1𝑤2𝑤3𝑡}+1^𝑖×𝑒^{𝑑𝑤2𝑤3𝑡}+1𝑒^{𝑑𝑤1𝑤3𝑡}+1𝑒^{𝑑𝑤1𝑤2𝑡}+1_𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤2𝑤3𝑡}^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤1𝑤3𝑡}^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤1𝑤2𝑡}.(2.2)(b) Type Λ^𝑖13 (for 𝑖=0,1,2,3): 𝐼Λ^𝑖13=∫^𝑋3𝜒𝑥₁𝜒𝑥₂𝜒𝑥₃𝑒^{(𝑤1𝑥1+𝑤2𝑥2+𝑤3𝑥3+𝑤1𝑤2𝑤3(∑3−𝑖𝑗=1𝑦𝑗))𝑡}𝑑𝑥₁𝑑𝑥₂𝑑𝑥₃∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑤3𝑥4𝑡}𝑑𝑥₄^𝑖=2(2.3)_3−𝑖𝑒^{𝑤1𝑤2𝑤3(∑3−𝑖𝑗=1𝑦𝑗)𝑡}𝑒^{𝑑𝑤1𝑤2𝑤3𝑡}+1^𝑖×𝑒^{𝑑𝑤1𝑡}+1𝑒^{𝑑𝑤2𝑡}+1𝑒^{𝑑𝑤3𝑡}+1_𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤1𝑡}^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤2𝑡}^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤3𝑡}.(2.4)(c-0) Type Λ⁰¹²: 𝐼Λ⁰¹²==_𝑋3𝜒𝑥8₁𝑒𝜒𝑥₂𝜒𝑥₃𝑒^{(𝑤1𝑥1+𝑤2𝑥2+𝑤3𝑤3+𝑤2𝑤3𝑦+𝑤1𝑤3𝑦+𝑤1𝑤2𝑦)𝑡}_{(𝑤2𝑤3+𝑤1𝑤3+𝑤1𝑤2)𝑦𝑡}𝑑𝑥₁𝑑𝑥₂𝑑𝑥₃(2.5)×𝑒^{𝑑𝑤1𝑡}+1𝑒^{𝑑𝑤2𝑡}+1𝑒^{𝑑𝑤3𝑡}+1_𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤1𝑡}^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤2𝑡}^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤3𝑡}.(2.6)(c-1) Type Λ¹¹²: 𝐼Λ¹¹²=∫^𝑋3𝜒𝑥₁𝜒𝑥₂𝜒𝑥₃𝑒^{(𝑤1𝑥1+𝑤2𝑥2+𝑤3𝑥3)𝑡}𝑑𝑥₁𝑑𝑥₂𝑑𝑥₃∫_𝑋3𝑒^{𝑑(𝑤2𝑤3𝑧1+𝑤1𝑤3𝑧2+𝑤1𝑤2𝑧3)𝑡}𝑑𝑧₁𝑧₂𝑧₃=𝑒(2.7)_{𝑑𝑤2𝑤3𝑡}+1𝑒^{𝑑𝑤1𝑤3𝑡}+1𝑒^{𝑑𝑤1𝑤2𝑡}+1×𝑒^{𝑑𝑤1𝑡}+1𝑒^{𝑑𝑤2𝑡}+1𝑒^{𝑑𝑤3𝑡}+1_𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤1𝑡}^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤2𝑡}^𝑑−1_𝑎=0(−1)^𝑎𝜒(𝑎)𝑒^{𝑎𝑤3𝑡}.(2.8)

All of the above 𝑝-adic integrals of various types are invariant under all permutations of 𝑤₁, 𝑤₂, 𝑤₃, 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, 𝑤₁, 𝑤₂, 𝑤₃ are all odd positive integers except for (a-0) and (c-0), where they are any positive integers. First, let's consider Type Λ^𝑖23, for each 𝑖=0,1,2,3. The following results can be easily obtained from (1.9) and (1.12):

(a-0)𝐼=Λ⁰²³_𝑋𝜒𝑥₁𝑒^{𝑤2𝑤3(𝑥1+𝑤1𝑦1)𝑡}𝑑𝑥₁_𝑋𝜒𝑥₂𝑒^{𝑤1𝑤3(𝑥2+𝑤2𝑦2)𝑡}𝑑𝑥₂_𝑋𝜒𝑥₃𝑒^{𝑤1𝑤2(𝑥3+𝑤3𝑦3)𝑡}𝑑𝑥₃=^∞_𝑘=0𝐸_𝑘,𝜒𝑤₁𝑦₁𝑘!𝑤²𝑤₃𝑡^𝑘^∞_𝑙=0𝐸_𝑙,𝜒𝑤₂𝑦₂𝑙!𝑤¹𝑤₃𝑡^𝑙^∞_𝑚=0𝐸_𝑚,𝜒𝑤₃𝑦₃𝑚!𝑤¹𝑤₂𝑡^𝑚=_∞_𝑛=0_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦₁𝐸_𝑙,𝜒𝑤₂𝑦₂𝐸_𝑚,𝜒𝑤₃𝑦₃𝑤^1𝑙+𝑚𝑤^2𝑘+𝑚𝑤^3𝑘+𝑙𝑛𝑡^𝑛!,(3.1) where the inner sum is over all nonnegative integers 𝑘, 𝑙, 𝑚 with 𝑘+𝑙+𝑚=𝑛 and 𝑘,𝑛𝑙,𝑚=𝑘!𝑛𝑙!!𝑚!.(3.2)(a-1) Here we write 𝐼(Λ¹²³) in two different ways: (1)𝐼=Λ¹²³_𝑋𝜒𝑥₁𝑒^{𝑤2𝑤3(𝑥1+𝑤1𝑦1)𝑡}𝑑𝑥₁_𝑋𝜒𝑥₂𝑒^{𝑤1𝑤3(𝑥2+𝑤2𝑦2)𝑡}𝑑𝑥₂∫_𝑋𝜒𝑥₃𝑒^{𝑤1𝑤2𝑥3𝑡}𝑑𝑥₃∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑤3𝑥4𝑡}𝑑𝑥₄=^∞_𝑘=0𝐸_𝑘,𝜒𝑤₁𝑦₁𝑤²𝑤₃𝑡^𝑘𝑘!^∞_𝑙=0𝐸_𝑙,𝜒𝑤₂𝑦₂𝑤¹𝑤₃𝑡^𝑙×𝑇𝑙!_𝑚𝑤₃𝑑−1,𝜒𝑤¹𝑤₂𝑡^𝑚=𝑚!(3.3)_∞_𝑛=0_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦₁𝐸_𝑙,𝜒𝑤₂𝑦₂𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^1𝑙+𝑚𝑤^2𝑘+𝑚𝑤^3𝑘+𝑙𝑛𝑡^𝑛!.(3.4)(2) Invoking (1.13), (3.3) can also be written as 𝐼=Λ¹²³_{𝑤3𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)_𝑋𝜒𝑥₁𝑒^{𝑤2𝑤3(𝑥1+𝑤1𝑦1)𝑡}𝑑𝑥₁_𝑋𝜒𝑥₂𝑒^{𝑤1𝑤3(𝑥2+𝑤2𝑦2+𝑤2/𝑤3𝑎)𝑡}𝑑𝑥₂=^{𝑤3𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)^∞_𝑘=0𝐸_𝑘,𝜒𝑤₁𝑦₁𝑤²𝑦₃𝑡^𝑘×𝑘!_∞_𝑙=0𝐸_𝑙,𝜒𝑤²𝑦₂+𝑤²𝑤₃𝑎𝑤¹𝑦₃𝑡^𝑙=𝑙!_∞_𝑛=0⎛⎜⎜⎝𝑤_𝑛3𝑛_𝑘=0𝑘𝑛𝐸^𝑘,𝜒𝑤₁𝑦₁^{𝑤3𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_{𝑛−𝑘,𝜒}𝑤²𝑦₂+𝑤²𝑤₃𝑎𝑤^1𝑛−𝑘𝑤^𝑘2⎞⎟⎟⎠𝑡_𝑛𝑛!.(3.5)(a-2) Here we write 𝐼(Λ²²³) in three different ways: (1)𝐼=Λ²²³_𝑋𝜒𝑥₁𝑒^{𝑤2𝑤3(𝑥1+𝑤1𝑦1)𝑡}𝑑𝑥₁∫_𝑋𝜒𝑥₂𝑒^{𝑤1𝑤3𝑥2𝑡}𝑑𝑥₂∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑤3𝑥4𝑡}𝑑𝑥₄∫_𝑋𝜒𝑥₃𝑒^{𝑤1𝑤2𝑥3𝑡}𝑑𝑥₃∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑤3𝑥4𝑡}𝑑𝑥₄=^∞_𝑘=0𝐸_𝑘,𝜒𝑤₁𝑦₁𝑤²𝑤₃𝑡^𝑘𝑘!^∞_𝑙=0𝑇_𝑙𝑤₂𝑑−1,𝜒𝑤¹𝑤₃𝑡^𝑙×𝑙!_∞_𝑚₌₀𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤¹𝑤₂𝑡^𝑚=𝑚!(3.6)_∞_𝑛=0_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁×𝑤𝑦₁𝑇_𝑙𝑤₂𝑑−1,𝜒₃𝑑−1,𝜒𝑤^1𝑙+𝑚𝑤^2𝑘+𝑚𝑤^3𝑘+𝑙𝑇^𝑚𝑡^𝑛𝑛!.(3.7)(2) Invoking (1.13), (3.6) can also be written as 𝐼=Λ²²³_{𝑤2𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)_𝑋𝜒𝑥₁𝑒^{𝑤2𝑤3(𝑥1+𝑤1𝑦1+(𝑤1/𝑤2)𝑎)𝑡}𝑑𝑥₁×∫^𝑋𝜒𝑥₃𝑒^{𝑤1𝑤2𝑥3𝑡}𝑑𝑥₃∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑤3𝑥4𝑡}𝑑𝑥₄=^{𝑤2𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)^∞_𝑘=0𝐸_𝑘,𝜒𝑤¹𝑦₁+𝑤¹𝑤₂𝑎𝑤²𝑤₃𝑡^𝑘×𝑘!_∞_𝑙=0𝑇_𝑙𝑤₃𝑑−1,𝜒𝑤¹𝑤₂𝑡^𝑙=𝑙!(3.8)_∞_𝑛=0⎛⎜⎜⎝𝑤_𝑛2𝑛_𝑘=0𝑘𝑛^{𝑤2𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_𝑘,𝜒𝑤¹𝑦₁+𝑤¹𝑤₂𝑎𝑇_𝑛−𝑘×𝑤₃𝑑−1,𝜒𝑤^1𝑛−𝑘𝑤^𝑘3⎞⎟⎟⎠𝑡_𝑛𝑛!.(3.9)(3) Invoking (1.13) once again, (3.8) can be written as 𝐼=Λ²²³_{𝑤2𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)^{𝑤3𝑑−1}_𝑏=0(−1)^𝑏𝜒(𝑏)_𝑋𝜒𝑥₁𝑒^{𝑤2𝑤3(𝑥1+𝑤1𝑦1+(𝑤1/𝑤2𝑎)+(𝑤1/𝑤3)𝑏)𝑡}𝑑𝑥₁=^{𝑤2𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)^{𝑤3𝑑−1}_𝑏=0(−1)^𝑏𝜒(𝑏)^∞_𝑛=0𝐸_𝑛,𝜒𝑤¹𝑦₁+𝑤¹𝑤₂𝑎+𝑤¹𝑤₃𝑏𝑤²𝑤₃𝑡^𝑛=𝑛!(3.10)_∞_𝑛=0⎛⎜⎜⎝𝑤₂𝑤₃^{𝑛𝑤2𝑑−1}_𝑤𝑎=0^3𝑑−1_𝑏=0(−1)^𝑎+𝑏𝜒(𝑎𝑏)𝐸_𝑛,𝜒𝑤¹𝑦₁+𝑤¹𝑤₂𝑎+𝑤¹𝑤₃𝑏⎞⎟⎟⎠𝑡^𝑛𝑛!.(3.11)(a-3)𝐼Λ³²³=∫^𝑋𝜒𝑥₁𝑒^{𝑤2𝑤3𝑥1𝑡}𝑑𝑥₁∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑤3𝑥4𝑡}𝑑𝑥₄×∫^𝑋𝜒𝑥₂𝑒^{𝑤1𝑤3𝑥2𝑡}𝑑𝑥₂∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑤3𝑥4𝑡}𝑑𝑥₄×∫^𝑋𝜒𝑥₃𝑒^{𝑤1𝑤2𝑥3𝑡}𝑑𝑥₃∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑤3𝑥4𝑡}𝑑𝑥₄=^∞_𝑘=0𝑇_𝑘𝑤₁𝑑−1,𝜒𝑤²𝑤₃𝑡^𝑘𝑘!^∞_𝑙=0𝑇_𝑙𝑤₂𝑑−1,𝜒𝑤¹𝑤₃𝑡^𝑙×𝑙!_∞_𝑚₌₀𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤¹𝑤₂𝑡^𝑚=𝑚!(3.12)_∞_𝑛=0_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝑇^𝑘𝑤₁𝑑−1,𝜒𝑇_𝑙𝑤₂𝑑−1,𝜒𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^1𝑙+𝑚𝑤^2𝑘+𝑚𝑤^3𝑘+𝑙𝑛𝑡(^𝑛3!..13)(b) For Type Λ^𝑖13 (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 𝑤₂𝑤₃, 𝑤₁𝑤₃, 𝑤₁𝑤₂, respectively, for 𝑤₁, 𝑤₂, 𝑤₃ in (2.1), this amounts to replacing 𝑡 by 𝑤₁𝑤₂𝑤₃𝑡 in (2.3). So, upon replacing 𝑤₁, 𝑤₂, 𝑤₃, respectively, by 𝑤₂𝑤₃, 𝑤₁𝑤₃, 𝑤₁𝑤₂ and dividing by (𝑤₁𝑤₂𝑤₃)^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 Λ^𝑖13 (𝑖=0,1,2,3).(c-0)𝐼Λ⁰¹²=_𝑋𝜒𝑥₁𝑒^{𝑤1(𝑥1+𝑤2𝑦)𝑡}𝑑𝑥₁_𝑋𝜒𝑥₂𝑒^{𝑤2(𝑥2+𝑤3𝑦)𝑡}𝑑𝑥₂_𝑋𝜒𝑥₃𝑒^{𝑤3(𝑥3+𝑤1𝑦)𝑡}𝑑𝑥₃=^∞_𝑘=0𝐸_𝑘,𝜒𝑤₂𝑦𝑘!𝑤¹𝑡^𝑘^∞_𝑙=0𝐸_𝑙,𝜒𝑤₃𝑦𝑙!𝑤²𝑡^𝑙^∞_𝑚=0𝐸_𝑚,𝜒𝑤₁𝑦𝑚!𝑤³𝑡^𝑚=_∞_𝑛=0_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₂𝑦𝐸_𝑙,𝜒𝑤₃𝑦𝐸_𝑚,𝜒𝑤₁𝑦𝑤^𝑘1𝑤^𝑙2𝑤^𝑚3𝑛𝑡^𝑛!.(3.14)(c-1)𝐼Λ¹¹²=∫^𝑋𝜒𝑥₁𝑒^{𝑤1𝑥1𝑡}𝑑𝑥₁∫_𝑋𝑒^{𝑑𝑤1𝑤2𝑧3𝑡}𝑑𝑧₃×∫^𝑋𝜒𝑥₂𝑒^{𝑤2𝑥2𝑡}𝑑𝑥₂∫_𝑋𝑒^{𝑑𝑤2𝑤3𝑧1𝑡}𝑑𝑧₁×∫^𝑋𝜒𝑥₃𝑒^{𝑤3𝑥3𝑡}𝑑𝑥₃∫_𝑋𝑒^{𝑑𝑤3𝑤1𝑧2𝑡}𝑑𝑧₂=^∞_𝑘=0𝑇_𝑘𝑤₂𝑑−1,𝜒𝑤¹𝑡^𝑘𝑘!^∞_𝑙=0𝑇_𝑙𝑤₃𝑑−1,𝜒𝑤²𝑡^𝑙𝑙!^∞_𝑚=0𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤³𝑡^𝑚=𝑚!_∞_𝑛=0_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝑇^𝑘𝑤₂𝑑−1,𝜒𝑇_𝑙𝑤₃𝑑−1,𝜒𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤^𝑘1𝑤^𝑙2𝑤^𝑚3𝑛𝑡^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 𝑤₁, 𝑤₂, 𝑤₃. So the corresponding expressions in Section 3 are also invariant under any permutation of 𝑤₁, 𝑤₂, 𝑤₃. 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 𝑤₁, 𝑤₂, 𝑤₃ 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:_𝑘+_{𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁=𝑦₁𝑇_𝑙𝑤₃𝑑−1,𝜒𝑇_𝑚𝑤₂𝑑−1,𝜒𝑤^1𝑙+𝑚𝑤^3𝑘+𝑚𝑤^2𝑘+𝑙(4.1)_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₂=𝑦₁𝑇_𝑙𝑤₁𝑑−1,𝜒𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^2𝑙+𝑚𝑤^1𝑘+𝑚𝑤^3𝑘+𝑙(4.2)_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₃𝑦₁𝑇_𝑙𝑤₂𝑑−1,𝜒𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤^3𝑙+𝑚𝑤^2𝑘+𝑚𝑤^1𝑘+𝑙.(4.3) But, by interchanging 𝑙 and 𝑚, 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:_𝑘+_{𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝑇^𝑘𝑤₂=𝑑−1,𝜒𝑇_𝑙𝑤₃𝑑−1,𝜒𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤^𝑘1𝑤^𝑙2𝑤^𝑚3(4.4)_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝑇^𝑘𝑤₃=𝑑−1,𝜒𝑇_𝑙𝑤₁𝑑−1,𝜒𝑇_𝑚𝑤₂𝑑−1,𝜒𝑤^𝑘2𝑤^𝑙3𝑤^𝑚1(4.5)_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝑇^𝑘𝑤₃=𝑑−1,𝜒𝑇_𝑙𝑤₂𝑑−1,𝜒𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤^𝑘1𝑤^𝑙3𝑤^𝑚2(4.6)_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝑇^𝑘𝑤₂𝑑−1,𝜒𝑇_𝑙𝑤₁𝑑−1,𝜒𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^𝑘3𝑤^𝑙2𝑤^𝑚1.(4.7) However, (4.4) and (4.5) are equal to (4.17), as we can see by applying the permutations 𝑘→𝑙, 𝑙→𝑚, 𝑚→𝑘 for (4.4) and 𝑘→𝑚, 𝑙→𝑘, 𝑚→𝑙 for (4.5). Similarly, we see that (4.6) and (4.7) are equal to (4.18), by applying permutations 𝑘→𝑙, 𝑙→𝑚, 𝑚→𝑘 for (4.6) and 𝑘→𝑚, 𝑙→𝑘, 𝑚→𝑙 for (4.7).

Theorem 4.1. Let 𝑤₁, 𝑤₂, 𝑤₃ be any positive integers. Then one has _𝑘+_{𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦₁𝐸_𝑙,𝜒𝑤₂𝑦₂𝐸_𝑚,𝜒𝑤₃𝑦₃𝑤^1𝑙+𝑚𝑤^2𝑘+𝑚𝑤^3𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦₁𝐸_𝑙,𝜒𝑤₃𝑦₂𝐸_𝑚,𝜒𝑤₂𝑦₃𝑤^1𝑙+𝑚𝑤^3𝑘+𝑚𝑤^2𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₂𝑦₁𝐸_𝑙,𝜒𝑤₁𝑦₂𝐸_𝑚,𝜒𝑤₃𝑦₃𝑤^2𝑙+𝑚𝑤^1𝑘+𝑚𝑤^3𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₂𝑦₁𝐸_𝑙,𝜒𝑤₃𝑦₂𝐸_𝑚,𝜒𝑤₁𝑦₃𝑤^2𝑙+𝑚𝑤^3𝑘+𝑚𝑤^1𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₃𝑦₁𝐸_𝑙,𝜒𝑤₁𝑦₂𝐸_𝑚,𝜒𝑤₂𝑦₃𝑤^3𝑙+𝑚𝑤^1𝑘+𝑚𝑤^2𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑙𝑛,m𝐸^𝑘,𝜒𝑤₃𝑦₁𝐸_𝑙,𝜒𝑤₂𝑦₂𝐸_𝑚,𝜒𝑤₁𝑦₃𝑤^3𝑙+𝑚𝑤^2𝑘+𝑚𝑤^1𝑘+𝑙.(4.8)

Theorem 4.2. Let 𝑤₁, 𝑤₂, 𝑤₃ be any odd positive integers. Then one has _𝑘+_{𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦₁𝐸_𝑙,𝜒𝑤₂𝑦₂𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^1𝑙+𝑚𝑤^2𝑘+𝑚𝑤^3𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦₁𝐸_𝑙,𝜒𝑤₃𝑦₂𝑇_𝑚𝑤₂𝑑−1,𝜒𝑤^1𝑙+𝑚𝑤^3𝑘+𝑚𝑤^2𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₂𝑦₁𝐸_𝑙,𝜒𝑤₁𝑦₂𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^2𝑙+𝑚𝑤^1𝑘+𝑚𝑤^3𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₂𝑦₁𝐸_𝑙,𝜒𝑤₃𝑦₂𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤^2𝑙+𝑚𝑤^3𝑘+𝑚𝑤^1𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₃𝑦₁𝐸_𝑙,𝜒𝑤₂𝑦₂𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤^3𝑙+𝑚𝑤^2𝑘+𝑚𝑤^1𝑘+𝑙=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₃𝑦₁𝐸_𝑙,𝜒𝑤₁𝑦₂𝑇_𝑚𝑤₂𝑑−1,𝜒𝑤^3𝑙+𝑚𝑤^1𝑘+𝑚𝑤^2𝑘+𝑙.(4.9)

Theorem 4.3. Let 𝑤₁, 𝑤₂, 𝑤₃ be any odd positive integers. Then one has 𝑤^𝑛1𝑛_𝑘=0𝑘𝑛𝐸^𝑘,𝜒𝑤₃𝑦₁^{𝑤1𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_{𝑛−𝑘,𝜒}𝑤²𝑦₂+𝑤²𝑤₁𝑎𝑤^3𝑛−𝑘𝑤^𝑘2=𝑤^𝑛1𝑛_𝑘=0𝑘𝑛𝐸^𝑘,𝜒𝑤₂𝑦₁^{𝑤1𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_{𝑛−𝑘,𝜒}𝑤³𝑦₂+𝑤³𝑤₁𝑎𝑤^2𝑛−𝑘𝑤^𝑘3=𝑤^𝑛2𝑛_𝑘=0𝑘𝑛𝐸^𝑘,𝜒𝑤₃𝑦₁^{𝑤2𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_{𝑛−𝑘,𝜒}𝑤¹𝑦₂+𝑤¹𝑤₂𝑎𝑤^3𝑛−𝑘𝑤^𝑘1=𝑤^𝑛2𝑛_𝑘=0𝑘𝑛𝐸^𝑘,𝜒𝑤₁𝑦₁^{𝑤2𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_{𝑛−𝑘,𝜒}𝑤³𝑦₂+𝑤³𝑤₂𝑎𝑤^1𝑛−𝑘𝑤^𝑘3=𝑤^𝑛3𝑛_𝑘=0𝑘𝑛𝐸^𝑘,𝜒𝑤₂𝑦₁^{𝑤3𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_{𝑛−𝑘,𝜒}𝑤¹𝑦₂+𝑤¹𝑤₃𝑎𝑤^2𝑛−𝑘𝑤^𝑘1=𝑤^𝑛3𝑛_𝑘=0𝑘𝑛𝐸^𝑘,𝜒𝑤₁𝑦₁^{𝑤3𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_{𝑛−𝑘,𝜒}𝑤²𝑦₂+𝑤²𝑤₃𝑎𝑤^1𝑛−𝑘𝑤^𝑘2.(4.10)

Theorem 4.4. Let 𝑤₁, 𝑤₂, 𝑤₃ be any odd positive integers. Then one has the following three symmetries in 𝑤₁, 𝑤₂, 𝑤₃: _𝑘+_{𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁=𝑦₁𝑇_𝑙𝑤₂𝑑−1,𝜒𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^1𝑙+𝑚𝑤^2𝑘+𝑚𝑤^3𝑘+𝑙(4.11)_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚E^𝑘,𝜒𝑤₂=𝑦₁𝑇_𝑙𝑤₃𝑑−1,𝜒𝑇_𝑚𝑤₁𝑑−1,𝜒𝑤^2𝑙+𝑚𝑤^3𝑘+𝑚𝑤^1𝑘+𝑙(4.12)_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₃𝑦₁𝑇_𝑙𝑤₁𝑑−1,𝜒𝑇_𝑚𝑤₂𝑑−1,𝜒𝑤^3𝑙+𝑚𝑤^1𝑘+𝑚𝑤^2𝑘+𝑙.(4.13)

Theorem 4.5. Let 𝑤₁, 𝑤₂, 𝑤₃ be any odd positive integers. Then one has 𝑤^𝑛1𝑛_𝑘=0𝑘𝑛^{𝑤1𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_𝑘,𝜒𝑤²𝑦₁+𝑤²𝑤₁𝑎𝑇_𝑛−𝑘𝑤₃𝑑−1,𝜒𝑤^2𝑛−𝑘𝑤^𝑘3=𝑤^𝑛1𝑛_𝑘=0𝑘𝑛^{𝑤1𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_𝑘,𝜒𝑤³𝑦₁+𝑤³𝑤₁𝑎𝑇_𝑛−𝑘𝑤₂𝑑−1,𝜒𝑤^3𝑛−𝑘𝑤^𝑘2=𝑤^𝑛2𝑛_𝑘=0𝑘𝑛^{𝑤2𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_𝑘,𝜒𝑤¹𝑦₁+𝑤¹𝑤₂𝑎𝑇_𝑛−𝑘𝑤₃𝑑−1,𝜒𝑤^1𝑛−𝑘𝑤^𝑘3=𝑤^𝑛2𝑛_𝑘=0𝑘𝑛^{𝑤2𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_𝑘,𝜒𝑤³𝑦₁+𝑤³𝑤₂𝑎𝑇_𝑛−𝑘𝑤₁𝑑−1,𝜒𝑤^3𝑛−𝑘𝑤^𝑘1=𝑤^𝑛3𝑛_𝑘=0𝑘𝑛^{𝑤3𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_𝑘,𝜒𝑤¹𝑦₁+𝑤¹𝑤₃𝑎𝑇_𝑛−𝑘𝑤₂𝑑−1,𝜒𝑤^1𝑛−𝑘𝑤^𝑘2=𝑤^𝑛3𝑛_𝑘=0𝑘𝑛^{𝑤3𝑑−1}_𝑎=0(−1)^𝑎𝜒(𝑎)𝐸_𝑘,𝜒𝑤²𝑦₁+𝑤²𝑤₃𝑎𝑇_𝑛−𝑘𝑤₁𝑑−1,𝜒𝑤^2𝑛−𝑘𝑤^k₁.(4.14)

Theorem 4.6. Let 𝑤₁, 𝑤₂, 𝑤₃ be any odd positive integers. Then one has the following three symmetries in 𝑤₁, 𝑤₂, 𝑤₃: 𝑤₁𝑤=₂𝑤^{𝑛𝑤1𝑑−1}₂_𝑤𝑎=0^2𝑑−1_𝑏=0(−1)^𝑎+𝑏𝜒(𝑎𝑏)𝐸_𝑛,𝜒𝑤³𝑦₁+𝑤𝑤³₁𝑎+𝑤𝑤³₂𝑏𝑤₃^{𝑛𝑤2𝑑−1}_𝑤𝑎=0^3𝑑−1_𝑏=0(−1)^𝑎+𝑏𝜒(𝑎𝑏)𝐸_𝑛,𝜒𝑤¹𝑦₁+𝑤¹𝑤₂𝑎+𝑤¹𝑤₃=𝑤₃𝑏𝑤₁^{𝑛𝑤3𝑑−1}_𝑤𝑎=0^1𝑑−1_𝑏=0(−1)^𝑎+𝑏𝜒(𝑎𝑏)𝐸_𝑛,𝜒𝑤²𝑦₁+𝑤²𝑤₃𝑎+𝑤²𝑤₁𝑏.(4.15)

Theorem 4.7. Let 𝑤₁, 𝑤₂, 𝑤₃ be any positive integers. Then one has the following two symmetries in 𝑤₁, 𝑤₂, 𝑤₃: _𝑘+_{𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦𝐸_𝑙,𝜒𝑤₂𝑦𝐸_𝑚,𝜒𝑤₃𝑦𝑤^𝑘3𝑤^𝑙1𝑤^𝑚2=_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝐸^𝑘,𝜒𝑤₁𝑦𝐸_𝑙,𝜒𝑤₃𝑦𝐸_𝑚,𝜒𝑤₂𝑦𝑤^𝑘2𝑤^𝑙1𝑤^𝑚3.(4.16)

Theorem 4.8. Let 𝑤₁, 𝑤₂, 𝑤₃ be any odd positive integers. Then one has the following two symmetries in 𝑤₁, 𝑤₂, 𝑤₃: _𝑘+_{𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝑇^𝑘𝑤₁𝑑−1,𝜒𝑇_𝑙𝑤₂𝑑−1,𝜒𝑇_𝑚𝑤₃𝑑−1,𝜒𝑤^𝑘3𝑤^𝑙1𝑤^𝑚2,(4.17)_{𝑘+𝑙+𝑚=𝑛}𝑘,𝑛𝑙,𝑚𝑇^𝑘𝑤₁𝑑−1,𝜒𝑇_𝑙𝑤₃𝑑−1,𝜒𝑇_𝑚𝑤₂𝑑−1,𝜒𝑤^𝑘2𝑤^𝑙1𝑤^𝑚3.(4.18)