Three-Term Recurrence Relations for Basic Hypergeometric Series

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

q超幾何級数の三項間漸化式

鈴木, 由佳

https://doi.org/10.15017/1931727

出版情報：Kyushu University, 2017, 博士（数理学）, 課程博士バージョン：

権利関係：

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Three-Term Recurrence Relations for Basic Hypergeometric Series

Yuka Suzuki

Graduate School of Mathematics Kyushu University

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To my family, Chiharu, Akiko, Kosuke, Chika, and Shizuru.

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Preface

Our main objective in this paper is to give explicit expressions for the coeﬃcients of three- term recurrence relations for the 2ϕ1 basic hypergeometric series. These expressions includeq- analogues of Ebisu’s (2012) results on three-term recurrence relations for the₂F₁hypergeometric series. Also, we give generalizations of Gasper’s (1981) transformation and summation formulas for basic hypergeometric series.

Basic hypergeometric series are series ∑

a_n with a_n+1/a_n a rational function of qⁿ for a fixed parameter q. They have a very significant property: Basic hypergeometric series satisfy certain identities for special values of their sums. There are many applications of this property.

For example, in combinatorial analysis some identities for special values of basic hypergeometric series enumerate partitions of positive integers. The author’s primary subject of interest is the derivation of identities for special values of r+1ϕr series, which are basic hypergeometric series with (r+ 1) +r parameters and one variable. Many of the known identities for special values of r+1ϕr series are special or limiting cases of Jackson’s (1921) identity for a terminating 8ϕ7

series. Since the 8ϕ7 series in the Jackson’s identity is a particular type (called very-well-poised balanced) of series, the special and limiting cases are also particular type of series. The author would like to develop another derivation of identities for special values of r+1ϕr series.

The₂ϕ₁basic hypergeometric series is aq-analogue of the₂F₁hypergeometric series. The fact that₂F₁series and₂ϕ₁ series each satisfy three-term recurrence relations is known: Gauss (1812) obtained fifteen fundamental three-term recurrence relations for 2F1 series. Heine (1847) obtained q-analogues of Gauss’s fifteen relations and also obtained another type of fundamental three-term recurrence relations for ₂ϕ₁ series in which two ₂ϕ₁ series are evaluated at x and the other one is evaluated at qx. From Gauss and Heine’s relations we can verify that the fact, because these relations can be iterated. Ebisu (2012) obtained explicit expressions for the coeﬃcients of certain three-term recurrence relations for₂F₁ series, and using these expressions, he (2017) found many identities for special values of 2F1 series, which include not only previously known identities but also previously unknown identities. From this point of view, it can be expected that identities for special values of₂ϕ₁series are derived from three-term recurrence relations for 2ϕ1 series.

In this paper, for any integersk, l, m, andn, we consider the following three-term recurrence relation:

2ϕ₁

(aq^k, bq^l

cq^m ;q, xqⁿ )

=Q·2ϕ₁

(aq, bq cq ;q, x

)

+R·2ϕ₁ (a, b

c ;q, x )

.

This is aq-analogue of the three-term recurrence relation for2F1series considered by Ebisu (2012).

In Chapter 1, we consider the above three-term recurrence relation under the restriction that n = 0. Using a q-diﬀerential equation, contiguity operators, Heine’s (1847) q-Euler transfor- mation formula, and Gasper’s (1981) transformation and summation formulas for basic hypergeometric series, we give explicit expressions for Q and R for any integers k, l, and m. These expressions include q-analogues of Ebisu’s (2012) results on three-term recurrence relations for

2F1 series. In Chapter 2, we consider the above three-term recurrence relation without the restriction that n = 0. We first derive two generalizations of Gasper’s transformation formula

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used in Chapter 1. Then, in almost the same way described in Chapter 1, but using the gen- eralized formulas instead of Gasper’s transformation and summation formulas, we give explicit expressions for Q and R for any integers k, l, m, and n. Also, as corollaries of the generaliza- tions of Gasper’s transformation formula, we give generalizations of Gasper’s (1981) summation formulas for basic hypergeometric series.

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Acknowledgments

I would like to express my sincere gratitude to my supervisor Professor Hiroyuki Ochiai for warm encouragement and helpful advice. He recommended me a good book on hypergeometric functions when I was a senior in college. Since then, I have started to become interested in hypergeometric functions and started to learn about their properties. I would also like to express my gratitude to Professor Yuichiro Taguchi. He taught me how to read mathematics when I was a junior in college.

In addition, I am grateful to Professor Masatoshi Noumi. His insightful comments helped me meaningfully improve this paper. I am also indebted to Akihito Ebisu for helpful comments. He gave me an interesting book on basic hypergeometric series and applications, which influenced my study. I owe Naoya Yamaguchi a debt of gratitude for suggesting the main topic treated in this paper.

Furthermore, I would like to thank my friends from Kyushu University. Especially, thanks to Yuri Bito, Tomoko Kuwayama, Sachie Shiraichi, Sayaka Arimura, Tomoyuki Tamura, Naoya Yamaguchi, and Khongorzul Dorjgotov, I had a fulfilling study life.

Finally, I would like to express my sincere thanks to my father Chiharu Suzuki, my mother Akiko Suzuki, my brother Kosuke Suzuki, and my sisters Chika Suzuki and Shizuru Suzuki for longtime support and exciting encouragement.

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Three-Term Recurrence Relations

1.1 Introduction

In this chapter, we give explicit expressions for the coeﬃcients of three-term recurrence relations for the ₂ϕ₁ basic hypergeometric series. These expressions include q-analogues of Ebisu’s [7]

results for the 2F1 hypergeometric series.

The_r+1ϕ_r basic hypergeometric series is defined by

r+1ϕ_r

(a, b₁, . . . , b_r c₁, . . . , c_r ;q, x

)

=_r+1ϕ_r(a, b₁, . . . , b_r;c₁, . . . , c_r;q, x) :=

∑∞ i=0

(a)i(b1)i· · ·(br)i

(q)_i(c₁)_i· · ·(c_r)_ixⁱ,

where (a)_i denotes the q-shifted factorial defined by (a)_i = (a;q)_i := (a;q)_∞/(aqⁱ;q)_∞ with (a)_∞ = (a;q)_∞ := ∏_∞

j=0(1−aq^j). It is assumed that |q| < 1 and none of the denominator parameters c₁, . . . , c_r are 1 or any negative integer power ofq.

When a =q^α, b= q^β, and c =q^γ, the series 2ϕ1(a, b;c;q, x) is a q-analogue of the hyper- geometric series2F1(α, β;γ;x). It is known that for any triples of integers (k, l, m) and (k^′, l^′, m^′), the three hypergeometric series

2F1

(α+k, β+l γ+m ;x

) , 2F1

(α+k^′, β+l^′ γ+m^′ ;x

) , 2F1

(α, β γ ;x

)

satisfy a linear relation with coeﬃcients that are rational functions of α, β, γ, and x. Such a relation is called a “three-term recurrence relation for ₂F₁ series.” Gauss obtained three-term recurrence relations for 2F1 series in the cases

(k, l, m),(k^′, l^′, m^′)∈ {(1,0,0),(−1,0,0),(0,1,0),(0,−1,0),(0,0,1),(0,0,−1)}, where (k, l, m)̸= (k^′, l^′, m^′). Thus, there are(₆

2

)= 15 pairs of (k, l, m) and (k^′, l^′, m^′) for which Gauss obtained three-term recurrence relations (see [14, Chapter 4, p. 71] for the fifteen relations obtained by Gauss). Ebisu [7] considered three-term recurrence relations for 2F1 series in the cases (k, l, m)∈Z³ and (k^′, l^′, m^′) = (1,1,1) of the following form:

2F1

(α+k, β+l γ+m ;x

)

=P1·2F1

(α+ 1, β+ 1 γ+ 1 ;x

)

+P2·2F1

(α, β γ ;x

)

. (1.1.1)

He showed that the coeﬃcients P1 and P2 can each be expressed as a sum of products of two

2F₁ series.

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In analogy to the situation for₂F₁ series, for any triples of integers (k, l, m) and (k^′, l^′, m^′), the three basic hypergeometric series

2ϕ1

(aq^k, bq^l cq^m ;q, x

) , 2ϕ1

(aq^k^′, bq^l^′ cq^m^′ ;q, x

) , 2ϕ1

(a, b c ;q, x

)

satisfy a linear relation with coeﬃcients that are rational functions of a, b, c, q, and x. We call such a relation the “three-term recurrence relation for 2ϕ1 series.” The fifteen three-term recurrence relations for 2ϕ1 series corresponding to Gauss’s three-term recurrence relations for

2F₁ series were obtained by Heine [11, p. 290, Formulas 18–32].

In this chapter, we consider the following three-term recurrence relation for2ϕ1 series corresponding to (1.1.1):

2ϕ₁

)

=Q·2ϕ₁

(aq, bq cq ;q, x

)

+R·2ϕ₁ (a, b

c ;q, x )

. (1.1.2)

We show that the pair (Q, R) of rational functions of a, b, c, q, and x is uniquely determined by (k, l, m), and we give explicit expressions for Qand R. It is important to note that using these explicit expressions, we are able to obtain the more general three-term recurrence relations

2ϕ₁

)

=Q^′·2ϕ₁

(aq^k^′, bq^l^′ cq^m^′ ;q, x

)

+R^′·2ϕ₁ (a, b

c ;q, x )

by eliminating₂ϕ₁(aq, bq;cq;q, x) from (1.1.2) for (k, l, m) and (k^′, l^′, m^′).

Throughout this chapter, unless explicitly stated otherwise, we assume that a, b, c, a

b, c a, c

b ∈/ q^Z∪ {0}. (1.1.3)

1.2 Main results

Here we present main results of this chapter. Note that without loss of generality, it is suﬃcient to consider (1.1.2) for only cases satisfyingk≤l, because the series₂ϕ₁(a, b;c;q, x) is symmetric with respect to the exchange ofa andb.

The following lemma asserts the uniqueness of the pair (Q, R) satisfying (1.1.2).

Lemma 1. For any triple of integers (k, l, m), the pair (Q, R) of rational functions of a, b, c, q, and x satisfying (1.1.2)is uniquely determined by (k, l, m).

Employing the method due to Vid¯unas [17, Section 3], we prove this lemma.

The following theorem gives explicit expressions forQ and R.

Theorem 2. For any integers k, l, and m, with k≤l, the coeﬃcients of the three-term recur- rence relation (1.1.2)can be expressed as

Q=Q(k, l, m) =−(1−a)(1−b)c (q−c)(1−c)

x¹⁻^max^{^m,0^}

(abqx/c)_max_{_k+l₋_m,0_}−₁P (k, l

m ;a, b c ;q, x

) , R=R(k, l, m) =− x⁻^max^{^m⁻^1,0^}

(abqx/c)_max_{_k+l₋_m₋_1,0_}P

(k−1, l−1

m−1 ;aq, bq cq ;q, x

) .

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Here, P is the polynomial in x defined by

P (k, l

m ;a, b c ;q, x

) :=











l−1

∑

j=0

(A_j −B_j₋_m)x^j, k+l−m≥0, m≥0,

l−∑m−1 j=0

(A_j+m−B_j)x^j, k+l−m≥0, m <0,

m∑−k−1 j=0

(Ae_j −Be_j₋_m )

x^j, k+l−m <0, m≥0,

−∑k−1 j=0

(Ae_j+m−Be_j )

x^j, k+l−m <0, m <0, where Aj =Bj =Aej =Bej := 0 for any negative integer j, and

Aj := −c(aq/c)k−m(bq/c)l−m

(q²/c)₋m−1

(c)_m₋_j₋₁

(q⁻^j)j(a)_k₋_j(b)_l₋_jq^m⁻^j⁻¹4ϕ3

(q⁻^j, cq^m⁻^j⁻¹, a, b c, aq^k⁻^j, bq^l⁻^j ;q, q

) , B_j := (aq/c)j(bq/c)j

(q)_j(q²/c)_j ⁴ϕ₃

(q⁻^j, cq⁻^j⁻¹, cq^m⁻^k/a, cq^m⁻^l/b

cq^m, cq⁻^j/a, cq⁻^j/b ;q, q^k+l⁻^m+1 )

, Ae_j := (c)m

(a)_k(b)_l

(aq/c)j+k−m(bq/c)j+l−m

(q)_j(q²/c)_j₋_m ⁴ϕ₃

( q⁻^j, cq^m⁻^j⁻¹, c/a, c/b

c, cq^m⁻^k⁻^j/a, cq^m⁻^l⁻^j/b;q, q^m⁻^k⁻^l+1 )

, Be_j := (aq⁻^j)_j(bq⁻^j)_j

(q⁻^j)j(cq⁻^j⁻¹)j

q⁻^j₄ϕ₃

(q⁻^j, cq⁻^j⁻¹, aq^k, bq^l cq^m, aq⁻^j, bq⁻^j ;q, q

)

for any non-negative integer j.

The following lemma asserts thatP can be expressed as a sum of products of two 2ϕ1 series.

Lemma 3. Whenk≤l, the polynomialP defined in Theorem 2 can be rewritten in the following form:

P (k, l

m ;a, b c ;q, x

)

=











x⁻^min^{^m,0^}

∑∞ j=0

Ajx^j −x^max^{^m,0^}

∑∞ j=0

Bjx^j, k+l−m≥0, x⁻^min^{^m,0^}

∑∞ j=0

Aejx^j −x^max^{^m,0^}

∑∞ j=0

Bejx^j, k+l−m <0.

(1.2.1)

Moreover, each of the infinite series can be written as a product of two 2ϕ1 series as follows:

∑∞ j=0

A_jx^j = (aq/c)_k₋_m(bq/c)_l₋_m(c)_m (q²/c)₋m(a)k(b)l 2ϕ₁

(q^1−k/a, q^1−l/b

q²⁻^m/c ;q, abq^k+l−m

c x

)

2ϕ₁ (a, b

c ;q, x )

,

∑∞ j=0

B_jx^j =₂ϕ₁

(cq^m⁻^k/a, cq^m⁻^l/b

cq^m ;q, abq^k+l⁻^m

c x

)

2ϕ₁

(aq/c, bq/c q²/c ;q, x

) ,

∑∞ j=0

Ae_jx^j = (aq/c)_k₋_m(bq/c)_l₋_m(c)_m (q²/c)₋_m(a)_k(b)_l ²ϕ₁

(aq^k+1⁻^m/c, bq^l+1⁻^m/c q²⁻^m/c ;q, x

)

2ϕ₁

(c/a, c/b c ;q, ab

c x )

,

∑∞ j=0

Bejx^j =2ϕ1

)

2ϕ1

(q/a, q/b q²/c ;q, ab

c x )

.

The product of−c(a)_k(b)_l/{q(c/q)m+1} and this expression for P (as a sum of products of two ₂ϕ₁ series) is a q-analogue of the expression for q₀(x) given in [7, p. 263, Theorem 3.7].

Although it is not stated in [7], the expression forq0(x) can be written in the form of a multiple of a terminating 4F3 series.

As a corollary of Theorem 2, we obtain the following relation betweenQand R.

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Corollary 4. The coeﬃcients of the three-term recurrence relation (1.1.2) satisfy Q(k−1, l−1, m−1)

(a,b,c)7→(aq,bq,cq)

= (1−aq)(1−bq)x(c−abqx)

(1−c)(1−cq) R(k, l, m).

Note that the number of terms with degree j in the polynomial P defined in Theorem 2 increases as jgrows larger, because, in general,4ϕ3(q⁻^j, b1, b2, b3;c1, c2, c3;q, x) is a sum ofj+ 1 terms for any non-negative integerj. Now, we give an alternative expression forP in which the number of terms with degreej decreases as j grows larger.

The following proposition provides an alternative expression for P.

Proposition 5. For any integersk, l, andm, withk≤l, the polynomialP defined in Theorem 2 can be expressed as follows:

P (k, l

m ;a, b c ;q, x

)

=µ











l−1

∑

j=0

(Cj−Dj+k−l)x^l⁻¹⁻^j, k+l−m≥0, m≥0,

l−∑m−1 j=0

(Cj −Dj+k−l)x^l⁻^m⁻¹⁻^j, k+l−m≥0, m <0,

m∑−k−1 j=0

(Ce_j−De_j+k₋_l )

x^m⁻^k⁻¹⁻^j, k+l−m <0, m≥0,

−∑k−1 j=0

(Ce_j−De_j+k₋_l )

x⁻^k⁻¹⁻^j, k+l−m <0, m <0,

where Dj =Dej := 0 for any negative integer j, Cj :=µ1

(b)j(bq/c)j

(q)_j(bq/a)_j (cq

ab )_j

4ϕ3

(q⁻^j, aq⁻^j/b, q¹⁻^l/b, cq^m⁻^l/b aq^k⁻^l+1/b, q¹⁻^j/b, cq⁻^j/b ;q, q

) , D_j :=µ₂(q¹⁻^k/a)_j(cq^m⁻^k/a)_j

(q)_j(bq^l⁻^k+1/a)_j q^j₄ϕ₃

( q⁻^j, aq^k⁻^l⁻^j/b, a, aq/c

aq/b, aq^k⁻^j, aq^k⁻^m⁻^j+1/c;q, q^k+l⁻^m+1 )

, Cej :=µ1

(aq^k)_j(aq^k⁻^m+1/c)_j (q)j(aq^k⁻^l+1/b)j

(cq^m⁻^k⁻^l+1 ab

)j 4ϕ3

( q⁻^j, bq^l⁻^k⁻^j/a, q/a, c/a bq/a, q¹⁻^k⁻^j/a, cq^m⁻^k⁻^j/a;q, q

) , Dej :=µ2

(q/b)j(c/b)j

(q)_j(aq/b)_j q^j4ϕ3

(q⁻^j, bq⁻^j/a, bq^l, bq^l⁻^m+1/c

bq^l⁻^k+1/a, bq⁻^j, bq¹⁻^j/c ;q, q^m⁻^k⁻^l+1 )

, with

µ₁ :=a^m⁻^k⁻^l ( c

ab )k

q^k(m⁻^k⁻^l+1)(a)_k(aq/c)_k₋_m (aq/b)_k₋_l , µ2 :=b^m⁻^k⁻^l

( c ab

)l

q^l(m⁻^k⁻^l+1)(b)_l(bq/c)_l₋_m (bq/a)_l₋_k , for any non-negative integer j, and

µ:= (−1)^k+l⁻^m⁻^1+Mq^{^k(k⁻^1)+l(l⁻¹⁾⁻^m(m⁻^1)+M^(M⁻¹⁾^}^/2 (q−c)a^kb^l (b−a)c^m

(ab c

)_M (c)_m (a)_k(b)_l with M := max{k+l−m,0}.

Both the expressions for P given in Theorem 2 and Proposition 5 are useful in the con- struction of an algorithm for calculating P. More precisely, Theorem 2 is useful for calculating the coeﬃcients of lower-degree terms in P, whereas Proposition 5 is useful for calculating the coeﬃcients of higher-degree terms inP.

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1.3 Three-term recurrence relations for

₂

ϕ

₁

series

In this section, we prove the results presented in the previous section. In Section 1.3.1, employing the method due to Vid¯unas [17, Section 3], we prove Lemma 1. In Sections 1.3.2–1.3.7, we prove Theorem 2 and Lemma 3 in the following way: In Section 1.3.2, we introduce a series 2ϕe1 and rewrite the three-term recurrence relation (1.1.2) into a three-term recurrence relation for 2ϕe1

with coeﬃcientsQeandR. Then, comparing these two three-term recurrence relations, we obtaine expressions for Q and R in terms of Qe and R, respectively. In Section 1.3.3, we introduce foure local solutions yi (i = 1,2,3,4) of a q-diﬀerential equation Eq(a, b, c) defined by Ly(x) = 0, where

L:= (1−T_q)(1−cq⁻¹T_q)−x(1−aT_q)(1−bT_q)

with T_q :x7→ qx. Also, in Section 1.3.4, we introduce six q-diﬀerential operators called “conti- guity operators,” and combining these operators, we obtain a q-diﬀerential operator θ(k, l, m).

Then, in Section 1.3.5, operating θ(k, l, m) onyi’s, and using Lemma 1, we obtain linear equations for Qe and R. Solving these equations, we express eache Qe and Re as a ratio of infinite series defined as a sum of products of y_i’s. In order to obtain more explicit expressions for Qe and R, in Section 1.3.6, we introduce Heine’se q-Euler transformation formula and Gasper’s transformation and summation formulas for basic hypergeometric series. In Section 1.3.7, using the Gasper’s formulas, we prove Lemma 3, and using the Heine’s formula and Lemma 3, we obtain expressions for Qe and Re in terms of P and thereby complete the proof of Theorem 2.

In Section 1.3.8, we prove Proposition 5. We first define a function Pe and obtain an expression for Q in terms of Pe that diﬀers from the expression in terms of P given in Theorem 2. Then, employing the uniqueness of Q proved in Section 1.3.1, by comparing these expressions for Q, we are able to complete the proof of Proposition 5. In Section 1.3.9, by using Theorem 2, we prove Corollary 4.

1.3.1 Uniqueness of the pair (Q, R)

We prove Lemma 1. For this purpose, we use the following summation formula, which was independently discovered by Bailey [3] and Daum [5]:

2ϕ₁ (a, b

aq/b;q, −q b

)

= (−q)_∞(aq;q²)_∞(aq²/b²;q²)_∞

(−q/b)_∞(aq/b)_∞ , |q|<min{1,|b|}. (1.3.1) This is a q-analogue of Kummer’s summation formula

2F₁

( α, β α+ 1−β;−1

)

= Γ(1 + (α/2))Γ(α+ 1−β) Γ(1 +α)Γ((α/2) + 1−β). We also use the three-term recurrence relation

2ϕ1(aq, b, cq) =(1−b)(c−abx)

c−b ²ϕ1(aq, bq, cq)−b(1−c)

c−b ²ϕ1(a, b, c). (1.3.2) This follows from Heine’s [11, p. 290, Formula 24 and p. 292, Formulas 43 and 48] three-term recurrence relations

( 1− c

a )

2ϕ₁(aq⁻¹, b, c)−( 1−c

b )

2ϕ₁(a, bq⁻¹, c) = c b

( 1− b

a ) (

1−ab cqx

)

2ϕ₁(a, b, c),

2ϕ1(aq, b, c)−2ϕ1(a, b, c) = a(1−b)x

1−c ²ϕ1(aq, bq, cq),

2ϕ1(aq, bq⁻¹, c)−2ϕ1(a, b, c) = (aq−b)x

(1−c)q ²ϕ1(aq, b, cq).

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Here, ₂ϕ₁(a, b, c) denotes₂ϕ₁(a, b;c;q, x).

In order to prove by contradiction, let us assume that there are two distinct pairs of rational functions (Q₁, R₁) and (Q₂, R₂) satisfying (1.1.2). Then, we have

(Q1−Q2)·2ϕ1(aq, bq, cq) = (R2−R1)·2ϕ1(a, b, c).

This implies that 2ϕ1(aq, bq, cq)/2ϕ1(a, b, c) is a rational function ofa, b, c, q, andx. Therefore, from (1.3.2), we find that the functiong(a, b;c;q, x) defined by

g (a, b

c ;q, x )

:=2ϕ1

(aq, b cq ;q, x

) /

2ϕ1

(a, b c ;q, x

)

is also a rational function of a, b, c, q, and x. However, from (1.3.1), we have g

(a, b

aq/b;q,−q b

)

=₂ϕ₁

(aq, b

aq²/b;q, −q b

) /

2ϕ₁ (a, b

aq/b;q, −q b

)

= (

1−aq b

)(aq²;q²)_∞(aq³/b²;q²)_∞ (aq;q²)_∞(aq²/b²;q²)_∞ .

It turns out thatg(a, b;aq/b;q,−q/b)(=:g(a)) is a rational function ofa, and at the same time, g(a) has unbounded set of poles. This is a contradiction. Thus Lemma 1 is proved.

1.3.2 Rewriting of the three-term recurrence relation

We introduce a series2ϕe1 and rewrite the three-term recurrence relation (1.1.2) into a three- term recurrence relation for ₂ϕe₁.

Let2ϕe1 be the series defined by

2ϕe₁ (a, b

c ;x )

:= (q)_∞(c)_∞ (a)_∞(b)_∞²ϕ₁

(a, b c ;q, x

) , and let us rewrite (1.1.2) as

2ϕe1

(aq^k, bq^l cq^m ;x

)

=Qe·2ϕe1

(aq, bq cq ;x

)

+Re·2ϕe1

(a, b c ;x

)

. (1.3.3)

Comparing (1.1.2) with (1.3.3), we find that Qand R can be expressed in terms ofQe and Re as Q= (cq)m−1

(aq)_k₋₁(bq)_l₋₁Q,e (1.3.4)

R= (c)_m (a)k(b)l

R.e (1.3.5)

Hence, we investigate Qe and R. Note that, from Lemma 1, the pair (e Q,e R) satisfying (1.3.3) ise uniquely determined by (k, l, m).

1.3.3 Local solutions of the q-diﬀerential equation We introduce four local solutions ofE_q(a, b, c).

Letyi (i= 1,2,3,4) be the functions defined by y1

(a, b c ;x

)

=y1(a, b;c;x) :=2ϕe1

(a, b c ;x

) , y2

(a, b c ;x

)

=y2(a, b;c;x) :=x¹⁻^γ2ϕe1

(aq/c, bq/c q²/c ;x

) ,

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y3

(a, b c ;x

)

=y3(a, b;c;x) :=a^γ⁻^α⁻^β+1x⁻^α2ϕe1

(a, aq/c aq/b ; cq

abx )

, y4

(a, b c ;x

)

=y4(a, b;c;x) :=b^γ⁻^α⁻^β+1x⁻^β2ϕe1

(b, bq/c bq/a ; cq

abx )

, where a=q^α, b=q^β, and c=q^γ.

Then, using the general theory of linear diﬀerence equations mentioned in [6, p. 62, Theo- rem 2.15], we are able to verify the following lemma:

Lemma 6. Under the assumption (1.1.3), y_i(a, b;c;x) (i= 1,2) are linearly independent solu- tions around x = 0 of Eq(a, b, c), and yi(a, b;c;x) (i = 3,4) are linearly independent solutions around x=∞.

1.3.4 Contiguity operators

We introduce six q-diﬀerential operators called contiguity operators, and combining these operators, we obtain a q-diﬀerential operator θ(k, l, m), which sends the parametersa, b, and c toaq^k, bq^l, and cq^m, respectively.

LetH_j andB_j (j= 1,2,3) be the first-orderq-diﬀerential operators defined by H1(a, b, c) := 1−aTq,

H₂(a, b, c) := 1−bT_q,

H3(a, b, c) :={(c−a)(c−b)x}⁻¹{

c²+abx−(a+b)cx−c(c−abx)Tq

}, B₁(a, b, c) :={(q−a)(c−a)}⁻¹{

cq−a(q+c) +a²x+a(c−abx)T_q} , B₂(a, b, c) :={(q−b)(c−b)}⁻¹{

cq−b(q+c) +b²x+b(c−abx)T_q} , B3(a, b, c) := 1−cq⁻¹Tq.

The operators H_j (j = 1,2,3) increase the parametersa, b, and c by q times, respectively, while the operators Bj (j = 1,2,3) decrease these parameters byq times. Hence, we call these six operators “contiguity operators.” (See [11, p. 287, Formula 2 and p. 292, Formulas 42–

44] for the definitions of H₁, H₂, and B₃, and see [12, p. 46, Remark 2.1.4] for the method of deriving operatorsB₁, B₂, andH₃ fromH₁, H₂, andB₃, respectively.) In fact, performing direct calculations, we obtain the following lemma:

Lemma 7. Let us write y_i(a, b;c;x), y_i(aq^±¹, b;c;x), y_i(a, bq^±¹;c;x), and y_i(a, b;cq^±¹;x) as yi, yi(aq^±¹), yi(bq^±¹), andyi(cq^±¹), respectively. Then, for i= 1 and2, we have

H1(a, b, c)yi=yi(aq), B1(a, b, c)yi =yi(aq⁻¹), H₂(a, b, c)y_i=y_i(bq), B₂(a, b, c)y_i =y_i(bq⁻¹), H3(a, b, c)yi=yi(cq), B3(a, b, c)yi =yi(cq⁻¹).

Also, for i= 3 and 4, we have

H₁(a, b, c)y_i=−a·y_i(aq), B₁(a, b, c)y_i =−qa⁻¹·y_i(aq⁻¹), H₂(a, b, c)y_i=−b·y_i(bq), B₂(a, b, c)y_i =−qb⁻¹·y_i(bq⁻¹), H3(a, b, c)yi=−c⁻¹·yi(cq), B3(a, b, c)yi =−cq⁻¹·yi(cq⁻¹).

LetSq(a, b, c) denote the solution space ofEq(a, b, c) on a simply connected domain inC\ {0}. Then, from Lemmas 6 and 7, under the assumption (1.1.3), the mappings

H₁(a, b, c) :S_q(a, b, c)→S_q(aq, b, c), H₂(a, b, c) :S_q(a, b, c)→S_q(a, bq, c),

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H₃(a, b, c) :S_q(a, b, c)→S_q(a, b, cq)

are linear isomorphisms, and their inverse mappings are given by B₁(aq, b, c), B₂(a, bq, c), and B₃(a, b, cq), respectively. Therefore, combining H_j and B_j (j = 1,2,3), we obtain a linear isomorphism

θ(k, l, m) :Sq(a, b, c)→Sq(aq^k, bq^l, cq^m)

for any integers k, l, andm. Note that the definition ofθ(k, l, m) does not depend on the order of composition of contiguity operators. Here, we set the order as follows:

Sq(a, b, c)→ · · · →Sq(aq^k, b, c)→ · · · →Sq(aq^k, bq^l, c)→ · · · →Sq(aq^k, bq^l, cq^m).

1.3.5 Expressions for each Qe and Re as a ratio of infinite series

We express each Qe and Re as a ratio of infinite series defined as a sum of products of yi

(i= 1,2,3,4).

Put ∆ :=x⁻¹(1−Tq). By direct calculations, we obtain the following lemma:

Lemma 8. Let y_i(a, b, c) denote y_i(a, b;c;x). Then, we have

∆y_i(a, b, c) = {

yi(aq, bq, cq), i= 1,2,

−abc⁻¹·y_i(aq, bq, cq), i= 3,4.

Let Q(a, b, c, q, x) denote the field generated by a, b, c, q, and x over Q, and let us write the ring of polynomials in Tq over Q(a, b, c, q, x) as Q(a, b, c, q, x)[Tq], where Tq and x are not commutative becauseT_qx=qxT_q. Then, regardingθ(k, l, m) as an element ofQ(a, b, c, q, x)[T_q], we expressθ(k, l, m) as

θ(k, l, m) =p(T_q)·L+ ˆQ·T_q+ ˆR,

where ˆQ,Rˆ ∈ Q(a, b, c, q, x) and p(Tq) ∈Q(a, b, c, q, x)[Tq]. From Tq = 1−x∆, this expression can be rewritten as

θ(k, l, m) =p(Tq)·L−xQˆ·∆ + ( ˆQ+ ˆR). (1.3.6) From Lemmas 6–8 and the definition of y1, operating (1.3.6) ony1(a, b;c;x) gives

2ϕe₁

(aq^k, bq^l cq^m ;x

)

=−xQˆ·2ϕe₁

(aq, bq cq ;x

)

+ ( ˆQ+ ˆR)·2ϕe₁ (a, b

c ;x )

. (1.3.7) Since the pair (Q,e R) satisfying (1.3.3) is unique as a consequence of Lemma 1, comparing (1.3.7)e with (1.3.3), we find that −xQˆ =Qe and ˆQ+ ˆR=R. Namely, we havee

θ(k, l, m) =p(Tq)·L+Qe·∆ +R.e (1.3.8) From Lemmas 6–8, operating (1.3.8) ony_i(a, b;c;x) (i= 1,2,3,4) gives

yi

(aq^k, bq^l cq^m ;x

)

=Qe·yi

(aq, bq cq ;x

)

+Re·yi

(a, b c ;x

)

, i= 1,2, (1.3.9)

λ·yi

(aq^k, bq^l cq^m ;x

)

=−abc⁻¹Qe·yi

(aq, bq cq ;x

)

+Re·yi

(a, b c ;x

)

, i= 3,4, (1.3.10)

Three-Term Recurrence Relations for Basic Hypergeometric Series

q超幾何級数の三項間漸化式

Three-Term Recurrence Relations for Basic Hypergeometric Series

Preface

Acknowledgments

Contents

Chapter 1

Three-Term Recurrence Relations

1.1 Introduction

1.2 Main results

1.3 Three-term recurrence relations for

ϕ

series