A Note on the Generalized 𝑞-Bernoulli Measures with Weight 𝛼

Abstract

We discuss a new concept of the 𝑞-extension of Bernoulli measure. From those measures, we derive some interesting properties on the generalized 𝑞-Bernoulli numbers with weight 𝛼 attached to 𝜒.

1. Introduction

Let 𝑝 be a fixed prime number. Throughout this paper ℤ_𝑝, ℚ_𝑝, and ℂ_𝑝 will, respectively, denote the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, and the completion of algebraic closure of ℚ_𝑝. Let ℕ be the set of natural numbers and ℤ₊=ℕ∪{0}. Let 𝜈_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=𝑝^{−𝜈𝑝(𝑝)}=1/𝑝 (see [1–14]).

When we talk of 𝑞-extension, 𝑞 is variously considered as an indeterminate, a complex number 𝑞∈ℂ, or a 𝑝-adic number 𝑞∈ℂ_𝑝. Throughout this paper we assume that 𝑞∈ℂ_𝑝 with |1−𝑞|_𝑝<1, and we use the notation of 𝑞-number as [𝑥]_𝑞=1−𝑞^𝑥,1−𝑞 (1.1) (see [1–14]). Thus, we note that lim_𝑞→1[𝑥]_𝑞=𝑥.

In [2], Carlitz defined a set of numbers 𝜉_𝑘=𝜉_𝑘(𝑞) inductively by 𝜉₀=1,𝑞𝜉+1^𝑘−𝜉_𝑘=1,if𝑘=1,0,if𝑘>1, (1.2) with the usual convention of replacing 𝜉^𝑘 by 𝜉_𝑘.

These numbers are 𝑞-extension of ordinary Bernoulli numbers 𝐵_𝑘. But they do not remain finite when 𝑞=1. So he modified (1.2) as follows:𝛽_0,𝑞=1,𝑞𝑞𝛽+1^𝑘−𝛽_𝑘,𝑞=1,if𝑘=1,0,if𝑘>1, (1.3) with the usual convention of replacing 𝛽^𝑘 by 𝛽_𝑘,𝑞.

The numbers 𝛽_𝑘,𝑞 are called the 𝑘-th Carlitz 𝑞-Bernoulli numbers.

In [1], Carlitz also considered the extended Carlitz’s 𝑞-Bernoulli numbers as follows:𝛽^ℎ_0,𝑞=ℎ[ℎ]_𝑞,𝑞^ℎ𝑞𝛽^ℎ+1^𝑘−𝛽^ℎ_𝑘,𝑞=1,if𝑘=1,0,if𝑘>1, (1.4) with the usual convention of replacing (𝛽^ℎ)^𝑘 by 𝛽^ℎ_𝑘,𝑞.

Recently, Kim considered 𝑞-Bernoulli numbers, which are different extended Carlitz’s 𝑞-Bernoulli numbers, as follows: for 𝛼∈ℕ and 𝑛∈ℤ₊,𝛽^(𝛼)_0,𝑞=1,𝑞𝑞^𝛼𝛽^(𝛼)+1^𝑛−𝛽^(𝛼)_𝑛,𝑞=𝛼[𝛼]_𝑞,if𝑛=1,0,if𝑛>1, (1.5) with the usual convention of replacing (𝛽^(𝛼))^𝑘 by 𝛽^(𝛼)_𝑘,𝑞 (see [3]).

The numbers 𝛽^(𝛼)_𝑘,𝑞 are called the 𝑘-th 𝑞-Bernoulli numbers with weight 𝛼.

For fixed 𝑑∈ℤ₊ with (𝑝,𝑑)=1, we set 𝑋=𝑋_𝑑=lim_←𝑁ℤ𝑑𝑝^𝑁ℤ,𝑋₁=ℤ_𝑝,𝑋^∗=_{0<𝑎<𝑑𝑝𝑎,𝑝=1}𝑎+𝑑𝑝ℤ_𝑝,𝑎+𝑑𝑝^𝑁ℤ_𝑝=𝑥∈𝑋∣𝑥≡𝑎mod𝑑𝑝^𝑁, (1.6) where 𝑎∈ℤ satisfies the condition 0≤𝑎<𝑑𝑝^𝑁.

Let UD(ℤ_𝑝) be the space of uniformly differentiable functions on ℤ_𝑝. For 𝑓∈UD(ℤ_𝑝), the 𝑝-adic 𝑞-integral on ℤ_𝑝 is defined by Kim as follows: 𝐼_𝑞𝑓=_ℤ𝑝𝑓(𝑥)𝑑𝜇_𝑞(𝑥)=lim_𝑁→∞1𝑝^𝑁_𝑞^𝑝𝑁−1_𝑥=0𝑓(𝑥)𝑞^𝑥, (1.7) (see [3, 4, 15, 16]). By (1.5) and (1.7), the Witt’s formula for the 𝑞-Bernoulli numbers with weight 𝛼 is given by _ℤ𝑝[𝑥]^𝑛_𝑞𝛼𝑑𝜇_𝑞(𝑥)=𝛽^(𝛼)_𝑛,𝑞,where𝑛∈ℤ₊. (1.8)

The 𝑞-Bernoulli polynomials with weight 𝛼 are also defined by 𝛽^(𝛼)_𝑛,𝑞(𝑥)=^𝑛_𝑙=0𝑛𝑙[𝑥]_𝑞^𝑛−𝑙_𝛼𝑞^𝛼𝑙𝑥𝛽^(𝛼)_𝑙,𝑞. (1.9)

From (1.7), (1.8), and (1.9), we can derive the Witt’s formula for 𝛽^(𝛼)_𝑛,𝑞(𝑥) as follows: _ℤ𝑝𝑥+𝑦^𝑛_𝑞𝛼𝑑𝜇_𝑞𝑦=𝛽^(𝛼)_𝑛,𝑞(𝑥),where𝑛∈ℤ₊. (1.10)

For 𝑛∈ℤ₊ and 𝑑∈ℕ, the distribution relation for the 𝑞-Bernoulli polynomials with weight 𝛼 are known that 𝛽^(𝛼)_𝑛,𝑞(𝑥)=[𝑑]^𝑛_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝑞^𝑎𝛽^(𝛼)_{𝑛,𝑞𝑑}𝑥+𝑎𝑑, (1.11) (see [3]). Recently, several authors have studied the 𝑝-adic 𝑞-Euler and Bernoulli measures on ℤ_𝑝 (see [8, 9, 11, 13, 14]). The purpose of this paper is to construct 𝑝-adic 𝑞-Bernoulli distribution with weight 𝛼 (=𝑝-adic 𝑞-Bernoulli unbounded measure with weight 𝛼) on ℤ_𝑝 and to study their integral representations. Finally, we construct the generalized 𝑞-Bernoulli numbers with weight 𝛼 and investigate their properties related to 𝑝-adic 𝑞-𝐿-functions.

2. 𝑝-Adic 𝑞-Bernoulli Distribution with Weight 𝛼

Let 𝑋 be any compact-open subset of ℚ_𝑝, such as ℤ_𝑝 or ℤ^∗_𝑝. A 𝑝-adic distribution 𝜇 on 𝑋 is defined to be an additive map from the collection of compact open set in 𝑋 to ℚ_𝑝:𝜇^𝑛_𝑘=1𝑈_𝑘=^𝑛_𝑘=1𝜇𝑈_𝑘additivity, (2.1) where {𝑈₁,𝑈₂,…,𝑈_𝑛} is any collection of disjoint compact opensets in 𝑋.

The set ℤ_𝑝 has a topological basis of compact open sets of the form 𝑎+𝑝^𝑛ℤ_𝑝.

Consequently, if 𝑈 is any compact open subset of ℤ_𝑝, it can be written as a finite disjoint union of sets 𝑈=^𝑘_𝑗=1𝑎_𝑗+𝑝^𝑛ℤ_𝑝, (2.2) where 𝑛∈ℕ and 𝑎₁,𝑎₂,…,𝑎_𝑘∈ℤ with 0≤𝑎_𝑖<𝑝^𝑛 for 𝑖=1,2,…𝑘.

Indeed, the 𝑝-adic ball 𝑎+𝑝^𝑛ℤ_𝑝 can be represented as the union of smaller balls 𝑎+𝑝^𝑛ℤ_𝑝=^𝑝−1_𝑏=0𝑎+𝑏𝑝^𝑛+𝑝^𝑛+1ℤ_𝑝. (2.3)

Lemma 2.1. Every map 𝜇 from the collection of compact-open sets in 𝑋 to ℚ_𝑝 for which 𝜇𝑎+𝑝^𝑁ℤ_𝑝=^𝑝−1_𝑏=0𝑎+𝑏𝑝^𝑁+𝑑𝑝^𝑁+1ℤ_𝑝 (2.4) holds whenever 𝑎+𝑝^𝑁ℤ_𝑝⊂𝑋, extends to a 𝑝-adic distribution (=𝑝-adic unbounded measure) on 𝑋.

Now we define a map 𝜇^(𝛼)_𝑘,𝑞 on the balls in ℤ_𝑝 as follows:𝜇^(𝛼)_𝑘,𝑞𝑎+𝑝^𝑛ℤ_𝑝=𝑝^𝑛𝑘_𝑞𝛼𝑝^𝑛_𝑞𝑞^𝑎𝑓^(𝛼)_{𝑘,𝑞𝑝𝑛}{𝑎}_𝑛𝑝^𝑛, (2.5) where {𝑎}_𝑛 is the unique number in the set {0,1,2,…,𝑝^𝑛−1} such that {𝑎}_𝑛≡𝑎(mod𝑝^𝑛).

If 𝑎∈{0,1,2,…,𝑝^𝑛−1}, then^𝑝−1_𝑏=0𝜇^(𝛼)_𝑘,𝑞𝑎+𝑏𝑝^𝑛+𝑝^𝑛+1ℤ_𝑝=^𝑝−1_𝑏=0𝑝^𝑛+1𝑘_𝑞𝛼𝑝^𝑛+1_𝑞𝑞^{𝑎+𝑏𝑝𝑛}𝑓^(𝛼)_{𝑘,𝑞𝑝𝑛+1}𝑎+𝑏𝑝^𝑛𝑝^𝑛+1=𝑞^𝑎𝑝^𝑛𝑘_𝑞𝛼𝑝^𝑛_𝑞𝑝^𝑘_𝑞𝑝^𝑛𝛼𝑝_𝑞𝑝^𝑛𝑝−1_𝑏=0𝑞^𝑏𝑝𝑛𝑓_𝑞^(𝛼)_{𝑘,𝑝𝑛𝑝}𝑎/𝑝^𝑛+𝑏𝑝. (2.6)

From (2.6), we note that 𝜇^(𝛼)_𝑘,𝑞 is 𝑝-adic distribution on ℤ_𝑝 if and only if 𝑝^𝑘_𝑞𝑝^𝑛𝛼𝑝_𝑞𝑝^𝑛𝑝−1_𝑏=0𝑞^𝑏𝑝𝑛𝑓_𝑞^(𝛼)_{𝑘,𝑝𝑛𝑝}𝑎/𝑝^𝑛+𝑏𝑝=𝑓^(𝛼)_{𝑘,𝑞𝑝𝑛}𝑎𝑝^𝑛. (2.7)

Theorem 2.2. Let 𝛼∈ℕ and 𝑘∈ℤ₊. Then we see that 𝜇^(𝛼)_𝑘,𝑞(𝑎+𝑝^𝑛ℤ_𝑝) is 𝑝-adic distribution on ℤ_𝑝 if and only if 𝑝^𝑘_𝑞𝑝^𝑛𝛼𝑝_𝑞𝑝^𝑛𝑝−1_𝑏=0𝑞^𝑏𝑝𝑛𝑓_𝑞^(𝛼)_{𝑘,𝑝𝑛𝑝}𝑎/𝑝^𝑛+𝑏𝑝=𝑓^(𝛼)_{𝑘,𝑞𝑝𝑛}𝑎𝑝^𝑛. (2.8)
One sets 𝑓^(𝛼)_{𝑘,𝑞𝑝𝑛}(𝑥)=𝛽^(𝛼)_{𝑘,𝑞𝑝}^𝑛(𝑥). (2.9)
From (2.5) and (2.9), one gets 𝜇^(𝛼)_𝑘,𝑞𝑎+𝑝^𝑛ℤ_𝑝=𝑝^𝑛𝑘_𝑞𝛼𝑝^𝑛_𝑞𝑞^𝑎𝛽^(𝛼)_{𝑘,𝑞𝑝}^𝑛𝑎𝑝^𝑛. (2.10)

By (1.11), (2.10), and Theorem 2.2, we obtain the following theorem.

Theorem 2.3. Let 𝜇^(𝛼)_𝑘,𝑞 be given by 𝜇^(𝛼)_𝑘,𝑞𝑎+𝑑𝑝^𝑁ℤ_𝑝=𝑑𝑝^𝑁𝑘_𝑞𝛼𝑑𝑝^𝑁_𝑞𝑞^𝑎𝛽^(𝛼)_𝑘,𝑞^𝑁_𝑑𝑝𝑎𝑑𝑝^𝑁. (2.11) Then 𝜇^(𝛼)_𝑘,𝑞 extends to a ℚ(𝑞)-valued distribution on the compact open sets 𝑈⊂𝑋.
From (2.11), one notes that _𝑋𝑑𝜇^(𝛼)_𝑘,𝑞(𝑥)=lim_𝑁→∞^{𝑑𝑝𝑁−1}_𝑥=0𝜇^(𝛼)_𝑘,𝑞𝑥+𝑑𝑝^𝑁ℤ_𝑝=lim_𝑁→∞𝑑𝑝^𝑁𝑘_𝑞𝛼𝑑𝑝^𝑁_𝑞^{𝑑𝑝𝑁−1}_𝑎=0𝑞^𝑎𝛽^(𝛼)_𝑘,𝑞^𝑁_𝑑𝑝𝑎𝑑𝑝^𝑁. (2.12)
By (1.11) and (2.12), one gets _𝑋𝑑𝜇^(𝛼)_𝑘,𝑞(𝑥)=𝛽^(𝛼)_𝑘,𝑞. (2.13)

Therefore, we obtain the following theorem.

Theorem 2.4. For 𝛼∈ℕ and 𝑘∈ℤ₊, one has _𝑋𝑑𝜇^(𝛼)_𝑘,𝑞(𝑥)=𝛽^(𝛼)_𝑘,𝑞. (2.14)
Let 𝜒 be Dirichlet character with conductor 𝑑∈ℕ. Then one defines the generalized 𝑞-Bernoulli numbers attached to 𝜒 as follows: 𝛽^(𝛼)_{𝑛,𝜒,𝑞}=_𝑋𝜒(𝑥)[𝑥]^𝑛_𝑞𝛼𝑑𝜇_𝑞(𝑥)=[𝑑]^𝑛_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝑞^𝑎𝜒(𝑎)𝛽^(𝛼)_{𝑛,𝑞𝑑}𝑎𝑑. (2.15)
From (2.11) and (2.15), one can derive the following equation; _𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_𝑘,𝑞(𝑥)=lim_𝑁→∞^{𝑑𝑝𝑁−1}_𝑥=0𝜒(𝑥)𝜇^(𝛼)_𝑘,q𝑥+𝑑𝑝^𝑁ℤ_𝑝=lim_𝑁→∞𝑑𝑝^𝑁𝑘_𝑞𝛼𝑑𝑝^𝑁_𝑞^{𝑑𝑝𝑁−1}_𝑥=0𝜒(𝑥)𝑞^𝑥𝛽^(𝛼)_𝑘,𝑞^𝑁_𝑑𝑝𝑥𝑑𝑝^𝑁=[𝑑]^𝑘_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝑞^𝑎𝜒(𝑎)lim_𝑁→∞𝑝^𝑁𝑘_𝑞𝛼𝑑𝑝^𝑁_𝑞𝑑^𝑝𝑁−1_𝑥=0𝑞^𝑑𝑥𝛽_𝑘,𝑞^𝑁_𝑑𝑝(𝑎/𝑑)+𝑥𝑝^𝑁=[𝑑]^𝑘_𝑞𝛼[𝑑]_𝑞^𝑑−1_𝑎=0𝑞^𝑎𝜒(𝑎)𝛽^(𝛼)_{𝑘,𝑞𝑑}𝑎𝑑=𝛽^(𝛼)_{𝑘,𝜒,𝑞},_𝑝𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_𝑘,𝑞(𝑥)=lim_𝑁→∞𝑑𝑝^𝑁+1𝑘_𝑞𝛼𝑑𝑝^𝑁+1_𝑞^{𝑑𝑝𝑁−1}_𝑥=0𝜒𝑝𝑥𝑞^𝑝𝑥𝛽^(𝛼)_{𝑘,𝑞𝑑𝑝}^𝑁+1𝑝𝑥𝑑𝑝^𝑁+1=𝑝^𝑘_𝑞𝛼𝑝_𝑞[𝑑]^𝑘_𝑞𝑝𝛼[𝑑]_𝑞𝑝^𝑑−1_𝑎=0𝜒𝑝𝑎𝑞^𝑝𝑎lim_𝑁→∞𝑝^𝑁𝑘_{𝑞𝑑𝑝𝛼}𝑝^𝑁_𝑞𝑑𝑝^𝑝𝑁−1_𝑥=0𝑞^𝑝𝑑𝑥𝛽^(𝛼)_𝑘,𝑞^𝑁_𝑝𝑑𝑝𝑝(𝑥𝑑+𝑎)𝑝𝑑𝑝^𝑁=𝑝^𝑘_𝑞𝛼𝑝_𝑞[𝑑]^𝑘_𝑞𝛼𝑝[𝑑]_𝑞𝑝^𝑑−1_𝑎=0𝜒𝑝𝜒(𝑎)𝑞^𝑝𝑎𝛽^(𝛼)_{𝑘,𝑞𝑝𝑑}𝑎𝑑𝑝𝑝=𝜒^𝑘𝑞𝛼𝑝_𝑞𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝}. (2.16)
For 𝛽(≠1)∈𝑋^∗, one has _𝑝𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_{𝑘,𝑞1/𝛽}𝑝𝛽𝑥=𝜒𝛽𝑝^𝑘_{𝑞𝛼/𝛽}𝑝_𝑞1/𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝/𝛽},_𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_{𝑘,𝑞1/𝛽}1𝛽𝑥=𝜒𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞1/𝛽}. (2.17)

Therefore, we obtain the following theorem.

Theorem 2.5. For 𝛽(≠1)∈𝑋^∗, one has _𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_𝑘,𝑞(𝑥)=𝛽^(𝛼)_{𝑘,𝜒,𝑞},_𝑝𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_𝑘,𝑞(𝑥)𝑝𝑝=𝜒^𝑘𝑞𝛼𝑝_𝑞𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝},_𝑝𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_{𝑘,𝑞1/𝛽}𝑝𝛽𝑥=𝜒𝛽𝑝^𝑘_{𝑞𝛼/𝛽}𝑝_𝑞1/𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝/𝛽},_𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_{𝑘,𝑞1/𝛽}1𝛽𝑥=𝜒𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞1/𝛽}. (2.18) Define 𝜇^(𝛼)_{𝑘,𝛽,𝑞}(𝑈)=𝜇^(𝛼)_𝑘,𝑞(𝑈)−𝛽⁻¹𝛽^−1𝑘_𝑞𝛼𝛽⁻¹_𝑞𝜇^(𝛼)_{𝑘,𝑞1/𝛽}𝛽𝑈. (2.19) By a simple calculation, one gets _𝑋∗𝜒(𝑥)𝑑𝜇^(𝛼)_{𝑘,𝛽,𝑞}(𝑥)=_𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_𝑘,𝑞(𝑥)−𝛽⁻¹𝛽^−1𝑘_𝑞𝛼𝛽⁻¹_𝑞^𝑝𝑋𝜒(𝑥)𝜇^(𝛼)_{𝑘,𝑞1/𝛽}(𝑥)=𝛽^(𝛼)_{𝑘,𝜒,𝑞}𝑝𝑝−𝜒^𝑘𝑞𝛼𝑝_𝑞𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝},𝛽^−1𝑘_𝑞𝛼𝛽^−1𝑘_𝑞^𝑋∗𝜒(𝑥)𝑑𝜇^(𝛼)_{𝑘,𝑞1/𝛽}𝛽𝑥=1/𝛽^𝑘_𝑞𝛼1/𝛽_𝑞𝜒1𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞1/𝛽}𝑝−𝜒𝛽𝑝/𝛽^𝑘_𝑞𝛼𝑝/𝛽_𝑞𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝/𝛽}. (2.20) By (2.19) and (2.20), one gets _𝑋∗𝜒(𝑥)𝑑𝜇^(𝛼)_{𝑘,𝛽,𝑞}𝛽𝑥=_𝑋𝜒(𝑥)𝑑𝜇^(𝛼)_𝑘,𝑞(𝑥)−𝛽⁻¹𝛽^−1𝑘_𝑞𝛼𝛽⁻¹_𝑞^𝑝𝑋𝜒(𝑥)𝜇^(𝛼)_{𝑘,𝑞1/𝛽}=𝛽𝛽𝑥_{(𝛼)𝑘,𝜒,𝑞}𝑝𝑝−𝜒^𝑘𝑞𝛼𝑝_𝑞𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝}−1𝛽1/𝛽^𝑘_𝑞𝛼1/𝛽_𝑞𝜒1𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞1/𝛽}𝑝+𝜒𝛽𝑝/𝛽^𝑘_𝑞𝛼𝑝/𝛽_𝑞𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝/𝛽}. (2.21)
Now one defines the operator 𝜒^𝑦=𝜒^{𝑦,𝑘,𝛼∶𝑞} on 𝑓(𝑞) by 𝜒^𝑦𝑓𝑞=𝜒^{𝑦,𝑘,𝛼∶𝑞}𝑓𝑞=𝑦^𝑘_𝑞𝛼𝑦_𝑞𝜒𝑦𝑓𝑞^𝑦. (2.22)
Thus, by (2.22), one gets 𝜒^{𝑥,𝑘,𝛼∶𝑞}∘𝜒^{𝑦,𝑘,𝛼∶𝑞}𝑓𝑞=𝜒^{𝑥,𝑘,𝛼∶𝑞}𝑦^𝑘_𝑞𝛼𝑦_𝑞𝜒𝑦𝑓𝑞^𝑦=𝑦^𝑘_𝑞𝛼𝑦_𝑞𝜒𝑦𝜒(𝑥)𝑦^𝑘_𝑞𝛼𝑦𝑦_𝑞𝑦𝜒𝑦𝑓𝑞^𝑥𝑦=𝑥𝑦^𝑘_𝑞𝛼𝑥𝑦_𝑞𝜒𝑥𝑦𝑓𝑞^𝑥𝑦=𝜒^{𝑥𝑦,𝑘,𝛼∶𝑞}𝑓𝑞=𝜒^𝑥𝑦𝑓𝑞. (2.23) Let us define 𝜒^𝑥𝜒^𝑦=𝜒^{𝑥,𝑘,𝛼∶𝑞}∘𝜒^{𝑦,𝑘,𝛼∶𝑞}. Then one has 𝜒^𝑥𝜒^𝑦=𝜒^𝑥𝑦. (2.24) From the definition of 𝜒^𝑥, one can easily derive the following equation; 1−𝜒^𝑝11−𝛽𝑥^1/𝛽1=1−𝛽𝑥^1/𝛽−𝜒^𝑝+1𝛽𝑥^𝑝/𝛽. (2.25) Let 𝑓(𝑞)=𝛽^(𝛼)_{𝑘,𝜒,𝑞}. Then one gets 1−𝜒^𝑝11−𝛽𝑥^1/𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞}=𝛽^(𝛼)_{𝑘,𝜒,𝑞}−1𝛽1/𝛽^𝑘_𝑞𝛼1/𝛽_𝑞𝜒1𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞}−𝑝^𝑘_𝑞𝛼𝑝_𝑞𝜒𝑝𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝}+1𝛽𝑝/𝛽^𝑘_𝑞𝛼𝑝/𝛽_𝑞𝜒𝑝𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞𝑝/𝛽}. (2.26)
By (2.21) and (2.26), one obtains the following equation: _𝑋∗𝜒(𝑥)𝑑𝜇^(𝛼)_{𝑘,𝛽,𝑞}𝛽𝑥=1−𝜒^𝑝11−𝛽𝑥^1/𝛽𝛽^(𝛼)_{𝑘,𝜒,𝑞}, (2.27) where 𝛽(≠1)∈𝑋^∗.