The q-borel Sum of Divergent Basic Hypergeometric series r ϕ s (a;b;q,x)

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1 arxiv: v1 [math.ca] 14 Jun 2018 The q-borel Sum of Divergent Basic Hypergeometric series r ϕ s a;b;q,x Shunya Adachi Abstract We study the divergent basic hypergeometric series which is a q-analog of divergent basic hypergeometric series. It is a formal solution of a certain linear q-difference equation. In this paper, we give an actual solution which admits basic hypergeometric series as a q-gevrey asymptotic expansion using q-borel-laplace transform. This result implies a q-analog of the Stokes phenomenon. After that, we show that our result formally degenerate into the known result about classical hypergeometric series by letting q 1 0. Our results are generalization of Zhang [10] and Morita [8]. 1 Introduction We set q C and 0 < q < 1 throughout this paper. Basic hypergeometric series with the base q is defined as a rϕ s a;b;q,x = r ϕ s b ;q,x := n 0 a 1,a 2,...,a r ;q { n b 1,...,b s ;q n q;q n } 1+s rx 1 n q nn 1 2 n, 1.1 where x C, a = a 1,a 2,...,a r C r, b = b 1,...,b s C s and b 1,...,b s / q N. In this paper, we assume a 1 a 2 a r 0 and b 1 b 2 b r 0. Basic hypergeometric series 1.1 is a q-analog of the Hypergeometric series α rf s α;β;x = r F s β ;x := n 0 α 1 n α 2 n α r n x n, 1.2 β 1 n β s n n! where α = α 1,α 2,...,α r C r, β = β 1,...,β s and β 1,...,β s / Z 0. The radius of converge of the series 1.1 and 1.2 is, 1 or 0 according to whether r > s+1. r = s+1 or r < s+1. Below, we assume r > s+1, i.e., 1.1 is divergent. Mathematical Subject Classification 2010: 33D15, 39A13, 34M30 Key words: basic hypergeometric series, q-difference equation, divergent series solution, q-borel summability, q-stokes phenomenon. 1

2 Basic hypergeometric series 1.1 formally satisfies the following linear q-difference equation: [ r s x σ q 1+s r 1 a j σ q 1 σ q 1 b ] k q σ q yx = 0, 1.3 j=1 where σ q is the q-shift operator given by σ q yx = yqx. This equation 1.3 has a fundamental system of solutions around infinity: y i x = θ q a i x ai, a iq θ q x r b φ 1, a iq b 2,..., a 1q b s,0,...,0 r 1 a i a a 1,..., i a a i 1, i a i+1,..., a ;q, qb 1 b s, 1 i r. 1.4 i a r a 1 a r x In this paper, we construct the actual solution of 1.3 which admits 1.1 as a asymptotic expansion using q-borel-laplace transform. See Theorem 3.1. After that, we show a q-analog of Stokes phenomenon by expressing its holomorphic solution by linear combination of 1.4. The end ofthis paper, we consider the case of taking the limit q 1 0 intheorem 3.1. See Theorem 3.2. The motivation of this study comes from as follows: Ichinobe [6] obtained the Borel sum of divergent classical hypergeometric series. Theorem 1.1 Ichinobe. Let α = α 1,α 2,...,α r C r, β = β 1,β 2,...,β s C s. Assume α i α j / Z i j. Then r F s α;β;x is 1/r s 1-summable in any direction d such that d 0 mod 2π and its Borel sum fx is given by fx = C αβ r j=1 k=1 C αβ j x α j αj,1+α j β s+1 F r 1 ; 1k, 1+α j α j x where x Sπ,r s+1π, := {x C ; π argx < r s+1π/2}, α j C r 1 is the vector which is obtained by omitting the j-th component from α and Here we use the following abbreviations Γα = C αβ = Γβ Γα, C αβj = Γα jγ α j α j. 1.5 Γβ α j r Γα l, Γ α j α j = j=1 r l=1, l j Γα l α j. Since basic hypergeometric series 1.1 is a q-analog of hypergeometric series 1.2, whether there exists a q-analog of Theorem 1.1 is a natural question. This is dealt with by the theory of q-borel summability. It is a q-analog of the theory of Borel summability and studied by many authors cf. Ramis [9], Zhang [10, 11], Di Vizio-Zhang [2] and Dreyfus-Eloy [3]. Dreyfus[4] proved that every formal power series solution of a linear q-difference equation 2

3 with rational coefficients, we may construct a meromorphic solution of the same equation applying several q-borel and Laplace transformations of convenient orders and convenient direction. Our results are generalization of Zhang [10] and Morita [8]. Zhang studied divergent hypergeometric series 2 ϕ 0 which is the case of r = 2 and s = 0 in our result. Later, Morita considered 3 ϕ 1 which is the case of r = 3 and s = 1 in our result. The common point of their assumptions is that r and s satisfy r s = 2. Under this assumption, the coefficients of the resummation of basic hypergeometric series may be expressed using functions with the base q. However, assuming r s > 2, the plural base appear mixedly in the coefficients. This paper organized as follows. We fix our notions and review the theory of q-borel summability in Section 2. Our main results are given in Section 3. In sections 4 and 5, we give proofs of Theorem 3.1 and Lemma 4.1 respectively. A proof of Theorem 3.2 is given in Section 6. 2 Preliminary 2.1 Basic Notions Let us fix our notations. We denote by 0 n the n-vector 0,0,...,0. For n = 0,1,2,..., We set the Pochhammer symbol as { 1 n = 0, a n := aa+1 a+n 1 n 1 and the q-shift factorial is defined by n 1 a;q n := 1 aq j, a;q := j=0 1 aq j. For any a C, the infinite product a;q is convergent. We set a 1,a 2,...,a r ;q n := r j=1 a j;q n for n = 0,1,2,... or n =. We set the Jacobi theta function with the base q by θ q x := n Z q nn 1 2 x n, which is holomorphic on C\{0}. The theta function holds following equalities: j=0 θ q x = q, x, q/x;q, 2.1 θ q q n x = x n q nn 1 2 θ q x n Z, 2.2 θ q qx = θ qx 1 = θ q. 2.3 x x 3

4 For λ C, we set a q-spiral [λ;q] := λq Z = {λq n ; n Z}. From the equality 2.1, we see θ q λ = 0 x [λ;q]. x We remark that the q-spiral [λ;q] is a class of C /q Z. The q-gamma function Γ q x is defined by Γ q x := q;q q x ;q 1 q 1 x. The limit q 1 0 of Γ q x gives the gamma function cf. [5] 2.2 Review of q-borel Summability lim Γ qx = Γx. 2.4 q 1 0 We review the theory of q-borel summability cf. Zhang [10, 11] and Dreyfus and Eloy [3]. Let C[[x]] be the ring of formal power series of x and ˆfx = n 0 a nx n C[[x]]. We define a q-borel transform ˆB q;1 : C[[x]] C[[ξ]] by ˆB q;1ˆfξ := n 0 a n q nn 1 2 ξ n. We set MC,0 be the field of functions that are meromorphic on some punctured neighborhood of 0 in C. More explicitly f MC,0 if and only if there exists V, an open neighborhood of 0, such that f is analytic on V \{0}. Let [λ;q] C /q Z and f MC,0. It is said that f belongs to H [λ;q] q;1 if there exists a positive constant ε and a domain Ω C such that: m Z{x C ; x λq m < ε q m λ } Ω. The function f can be continued to an analytic function on Ω with q-exponential growth at infinity, which means that there exist positive constants L and M such that for any ξ Ω, the following holds: fx < Lθ q M x. We note that it does not depend on the choice of λ. For [λ;q] C /q Z and f H [λ;q] q;1 MC,0 as L [λ;q] 1 λ fξ d q ξ q;1 fx := 1 q 0 θ q ξ ξ x, we define a q-laplace transform L[λ;q] q;1 : H [λ;q] 4 = n Z fλq n θ q λqn x. q;1

5 Here, this transformation is given in Jackson integral cf. [5]. For sufficient small x, the function L [λ;q] q;1 fx has poles of order at most one that are contained in the q-spiral [ λ;q]. The q-laplace transform is the inverse transform of q-borel transform, i.e., the following is hold: Lemma 2.1 cf. Dreyfus and Eloy [3]. Let [λ;q] C /q Z and f be an entire function. Then, the followings hold: ˆB q;1 f H [q;λ] q;1, L [λ;q] q;1 ˆB q;1 f = f. In the end of this section, we give asymptotic properties of the q-borel-laplace transform. Let [λ;q] C /q Z, ˆfx = n 0 a nx n C[[x]] and f MC,0. Then we define f [λ;q] 1 ˆf, if for any positive numbers ε and R, there exist positive constants C and K such that for any N 1 and x {x C ; x < R}\ m Z{x C ; x+λq m < ε q m λ }, we have N 1 fx n=0 a n x n LM N q NN 1 2 x N. We define that ˆfx is [λ;q]-summable if there exists a function f MC,0 such that f [λ;q] 1 ˆf. Lemma 2.2 cf. Dreyfus and Eloy [3]. Let [λ;q] C /q Z and f C[[x]] with ˆB q;1 f H [λ;q] q;1. Then L[λ;q] q;1 ˆB q;1 f [λ;q] 1 f. We call L [λ;q] q;1 ˆB q;1 f is the [λ;q]-sum of ˆf. From this lemma and the definition of [λ; q]-summable, we get the following corollary. Corollary 2.1. Let [λ;q] C /q Z and ˆf C[[x]]. If ˆBq;1 ˆf H [λ;q] q;1 is hold, ˆf is [λ;q]- summable. 5

6 3 Main Results In this section, we state our main results about the resummation of the basic hypergeometric series which is given by 1.1, ˆfx = r ϕ s a;b;q,x = r ϕ s a b ;q,x. 3.1 We remark that ˆfx is the formal solution of the linear q-difference equation 1.3. Our results are stated as follows. Theorem 3.1. We put k = r s 1 and p = q k. Assuming a i /a j / q Z i j, basic hypergeometric series 3.1 is [λ;p]-summable for any λ C \[ 1 k ;q] and its [λ;p]-sum rf s a;b;λ;q,x = L [λ:p] p;1 ˆB p;1ˆfx is given by rf s a;b;λ;q,x = a 2,...,a r,b 1 /a 1,...,b s /a 1 ;q θ p pa k 1 x/λ θ q 1 1 k a 1 λ b 1,...,b s,a 2 /a 1,...,a r /a 1 ;q θ p px/λ θ q 1 1 k λ a1,a r ϕ 1 q/b 1,...,a 1 q/b s,0 k r 1 ;q; qb 1 b s a 1 q/a 2,...,a 1 q/a r a 1 a r x +idema 1 ;a 2,...,a r, 3.2 wherex C \[ λ;p] and satisfies qb 1 b 2 b s / a 1 a r x < 1. Here the symbol idema 1 ;a 2,...,a r means the sum of r 1 expressions which are obtained by interchanging a 1 with each a 2,...,a r in the preceding expression. r f s a;b;λ;q,x has simple poles on p-spiral [ λ;p]. This theorem is a q-analog of Theorem 1.1. In addition, this theorem shows a q-analog of the Stokes phenomenon. Indeed, r f s is an actual solution of 1.3 which has 3.1 for its asymptotic expansion and its value changes depending on the choice of [λ;p] C /p Z. Using the fundamental system of solutions around infinity 1.4, the [λ;p]-sum r f s can be expressed as where M j x,λ = C j T j x,λ and rf s a;b;λ;q,x = r j=0 M j x,λy j x, C j = b 1/a j,b 2 /a j,...,b s /a j ;q b 1,b 2,...,b s ;q T j x,λ = θ ppa k j x/λ θ p px/λ θ q 1 1 k a j λ θ q 1 1 k λ 1 i r, i j a i ;q a i /a j ;q, θ q x θ q a j x. The coefficients M j are called q-stokes coefficient. Unlike the case of differential equation, the value of Stokes coefficient M j changes continuously depending on λ C\[ 1 k ;q]. Next, we have the following theorem by letting q 1 0 in Theorem

7 Theorem 3.2. Let α = α 1,α 2,...,α r C r, β = β 1,β 2,...,β s C s and α i α j / Z i j. We assume λ C \ [ 1 k ;q]. For x C \ [ λ;p] and arg x < π, the following equality holds: lim q 1 0 r f s q α ;q β ;λ;q, 1k x 1 q k = C αβ r j=1 C αβ j x α j s+1 F r 1 αj,1+α j β 1+α j α j ; 1k x, 3.3 where q α = q α 1,q α 2,...,q αr, q β = q β 1,q β 2,...,q βs and C αβ and C αβ j are given by 1.5. Weremarkthatifq 1 0therighthandsideof 3.3formallyconvergesto r F s α;β;x. 4 Proof of Theorem 3.1 First, we show that ˆfx = r ϕ s a;b;q,x is [λ;p]-summable for any λ C \[ 1 k ;q]. Let gξ be the p-borel transform of ˆfx a 1,a 2,...,a r ;q { } n 1+s rp gξ = ˆB q;1ˆfξ = 1 n q nn 1 nn ξ n. b 1,...,b s ;q n q;q n n 0 Then, gξ is convergent in ξ < 1 and can be expressed by basic hypergeometric series: gξ = a 1,a 2,...,a r ;q n 1 k ξ n a 1,a 2,...,a r = r ϕ r 1 b n 0 1,...,b s ;q n q;q n b 1,b 2,...,b s,0,...,0 ;q; 1 k ξ. In order to show the analytic continuation for gξ, we use the following proposition. Proposition 4.1 Slater [7]. The analytic continuation of the basic hypergeometric series is given by a1,...,a r rϕ r 1 ;q,ξ = a 2,...,a r,b 1 /a 1,...,b r 1 /a 1 ;q b 1,...,b r 1 b 1,...,b r 1,a 2 /a 1,...,a r /a 1 ;q θ q a 1 ξ a1,a 1 q/b 1,...,a 1 q/b r 1 θ q ξ r ϕ r 1 ;q, qb 1, b r 1 a 1 q/a 2,...,a 1 q/a r a 1 a r ξ +idema 1 ;a 2,...,a r. 4.1 Let b s+1,...,b r 1 0 in 4.1, gξ is continued analytically to ξ C \ [ 1 k ;q] as follows: gξ = a 2,...,a r,b 1 /a 1,...,b s /a 1 ;q θ q 1 k 1 a 1 ξ b 1...,b s,a 2 /a 1,...a r /a 1 ;q θ q 1 k 1 ξ s+1 ϕ r 1 ã; b;q; 1 k qk+1 b 1 b s 4.2 a 1 k 1 a 2 a r ξ +idema 1 ;a 2,...,a r, 7

8 Where ã = a 1,a 1 q/b 1,...,a 1 q/b s C s+1 and b = a 1 q/a 2,...,a 1 q/a r C r 1. We note that the basic hypergeometric series s+1 ϕ r 1 in 4.2 is holomorphic on C. In order to obtain 4.2, we can use lim aq/b;q nb n = limb aqb aq 2 b aq n = a n q nn+1 2. b 0 b 0 Now, we shall show that ˆB p;1ˆfξ = gξ H [λ;p] p;1 for any λ C \ [ 1 k ;q]. The following lemma is hold. Lemma 4.1. For δ > 0, we set a domain Ω δ as Ω δ := C \ m Z{ ξ C ; ξ + 1 k 1 q m < δ q m }. Then gξ has a p-exponential growth at infinity in Ω δ for any δ > 0. A proof of this lemma will be given in Section 5. For any λ C \[ 1 k ;q], there exists positive constants δ and ε such that {ξ C ; ξ λp m < ε q m λ } Ω δ. m Z Therefore, we obtain gξ H [λ;p] p;1 for any λ C \ [ 1 k ;q]. From Corollary 2.1, we see that ˆfx = r ϕ s a;b;q,x is [λ;p]-summable for any λ C\[ 1 k ;q]. Next, we derive the [λ;p]-sum of ˆfx, which is given by p-laplace transform of gξ. For any λ C\[ 1 k ;q], let r f s a;b;q,x be the p-laplace transform of gξ Since and s+1ϕ r 1 are hold, we have θ q 1 k 1 a 1 λp m θ q 1 k 1 λp m ã; b;q; 1 k rf s a;b;q,x = L [λ;p] p;1 gx = gλp m θ m Z p λpm. x = θ q 1 k 1 a 1 λq km θ q 1 k 1 λq km q k+1 b 1 b s = a1 1 k a 2 a r λp m n 0 λp m θ p = x = 1 θ q 1 k 1 a 1 λ a km 1 θ q 1 k 1 λ, ã;q n p nn 1 2 b,q;q n m λ λ p mm 1 2 θ p, x x pqb 1 b s a 1 k 1 a 2 a r λp m rf s a;b;q,x = a 2,...,a r,b 1 /a 1,...,b s /a 1 ;q θ q 1 k 1 a 1 λ 1 b 1...,b s,a 2 /a 1,...a r /a 1 ;q θ q 1 k 1 λ θ p λ/x n ã;q n qb1 b s m λ b,q;q n 0 n a 1 k 1 a 2 a r λ a k 1 x p nn 1 +n mn+ mm idema 1 ;a 2,...,a r. m Z n, 8

9 Now, m Z m λ p nn 1 +n mn+ mm 1 λ a k 2 2 = 1x a k 1x = λ a k 1 x n m Z n θ p λ a k 1 x p m nm n 1 2 m n λ a k 1x is hold. Therefore we obtain From and rf s a;b;q,x = a 2,...,a r,b 1 /a 1,...,b s /a 1 ;q θ q 1 k 1 a 1 λ b 1...,b s,a 2 /a 1,...a r /a 1 ;q θ q 1 k 1 λ n ã;q n qb1 b s b,q;q n 0 n a 1 a 2 a r x n 0 we finish the proof. ã;q n b,q;q n +idema 1 ;a 2,...,a r. θ p λ/a k 1x θ p λ/x = θ ppa k 1x/λ θ p px/λ θ p λ/a k 1 x θ p λ/x n ã,0k qb1 b s = r ϕ r 1 a 1 a 2 a r x b ;q, qb 1 b s, a 1 a 2 a r x 5 Proof of Lemma 4.1 In this section, we give a proof of Lemma 4.1, that is, we show gξ Lθ p M ξ, ξ Ω δ, where L and M are positive constants. As easily seen, the following inequality is hold. θ p A ξ +θ p B ξ θ p C ξ, wherea, B > 0andC = A+B. Hence, showingthatthefirsttermof 4.2hasp-exponential growth at infinity in Ω δ is sufficient to prove Lemma 4.1. We put the first term of 4.2 as g 1 ξ. i.e. g 1 ξ := a 2,...,a r,b 1 /a 1,...,b s /a 1 ;q b 1...,b s,a 2 /a 1,...a r /a 1 ;q θ q 1 k 1 a 1 ξ θ q 1 k 1 ξ s+1 ϕ r 1 where ã = a 1,a 1 q/b 1,...,a 1 q/b s C s+1 and b = a 1 q/a 2,...,a 1 q/a r C r 1. 9 ã; b;q; 1 k qk+1 b 1 b s, a 1 k 1 a 2 a r ξ

10 We show that basic hypergeometric series s+1 ϕ r 1 has p-exponential growth. It holds that α;q n = 1 α 1 αq 1 αq n 1 1+ α n, a i q/a j ;q n = 1 a i q/a j 1 a i q 2 /a j 1 a i q n /a j n inf { 1 a iq k /a j } > 0 k N for any α C and 1 i,j r. Therefore we obtain s+1 ϕ r 1 ã; b;q; 1 k qk+1 b 1 b s ã;q n p nn+1 a1 1 k 2 qb 1 b s a 2 a r ξ b,q;q n 0 n a 1 k 1 a 2 a r ξ n p nn M ξ n 0 n p nn = θ p M ξ, M ξ n Z where M is a positive constant. Hence we have g 1 ξ C θ q 1 k 1 a 1 ξ θ q 1 k 1 ξ θ p M ξ. 5.1 Here C is a positive constant. Now, for any ξ Ω δ, there exists some positive integer n and ξ 0 Ω δ {ξ C ; p ξ 1} such that ξ = p n ξ 0. Substituting ξ = p n ξ 0 in 5.1, we have g 1 ξ θ q 1 k 1 a 1 ξ 0 θ q 1 k 1 ξ 0 M a1 k n p nn 1 2 θ p M ξ 0. Since holds, we have M a1 k n p nn 1 2 = θ p M a 1 k p n ξ 0 = θ p M a 1 k ξ 0 Now, the function of ξ 0 g 1 ξ θ q 1 k 1 a 1 ξ 0 θ q 1 k 1 ξ 0 θ q 1 k 1 a 1 ξ 0 θ q 1 k 1 ξ 0 1 θ p M a 1 k ξ 0 θ p M a 1 k ξ θ p M ξ 0 θ p M a 1 k ξ 0 θ p M a 1 k ξ. θ p M ξ 0 θ p M a 1 k ξ 0 is holomorphic in the compact set Ω δ {ξ C ; p ξ 1}. Hence there exists a positive constant L such that max θ q 1 k 1 a 1 ξ 0 θ p M ξ 0 Ω δ {ξ C ; p ξ 1} θ q 1 k 1 ξ 0 θ p M a 1 k ξ 0 L. 10 n

11 6 Proof of Theorem 3.2 A proof of Theorem 3.2 is obtained from the following proposition. Proposition 6.1 Askey [1]. For any x C arg x < π, we have θ q q β x lim q 1 0 θ q q α x = xα β 6.1 and lim q 1 0 θ p θ p p α x 1 q k 1 q kβ α = x β α. 6.2 p β x 1 q k Let us give a proof of Theorem 3.2. Substituting a = q α, b = q β and x = 1 k x/1 q k in 3.2, we have rf s q α ;q β ;λ;q, 1k x = Γ qβ 1 Γ q β s Γ q α 2 α 1 Γ q α r α 1 1 q k Γ q α 2 Γ q α r Γ q β 1 α 1 Γ q β s α 1 1 q kα 1 θ p α k x p θ p 1 k x p λ1 q k λ1 q k θ q 1 1 k q α 1 λ θ q 1 1 k λ r ϕ r 1 q α 1,q 1+α 1 β 1,...,q 1+α 1 β s,0 k q 1+α 1 α 2,...,q 1+α 1 α r +idemq α 1 ;q α 2,...,q αr, where γ = 1+β 1 + +β s α 1 + +α r. Since we have ;q;q γ1 qk 1 k x Γ q β 1 Γ q β s Γ q α 2 α 1 Γ q α r α 1 lim q 1 0 Γ q α 2 Γ q α r Γ q β 1 α 1 Γ q β s α 1 = Γβ 1 Γβ s Γα 2 α 1 Γα r α 1 Γα 2 Γα r Γβ 1 α 1 Γβ s α 1, θ q 1 1 k q α 1 λ lim = { 1 1 k λ } α 1, q 1 0 θ q 1 1 k λ θ p α k x p λ1 q lim k q 1 0 θ p 1 k x p lim q 1 0 r ϕ r 1 1 q kα1 = λ1 q k q α 1,q 1+α 1 β 1,...,q 1+α 1 β s,0 k q 1+α 1 α 2,...,q 1+α 1 α r { 1 kx λ} α1, ;q;q γ1 qk α1,1+α 1 β = 1 k s+1 F r 1 ; 1k x 1+α 1 α 1 x from 2.4, 6.1, 6.2 and respectively, we obtain 3.3. lim q 1 0 q α ;q n 1 q n = α n 11

12 References [1] R. Askey, The q-gamma and q-beta functions, Appl. Anal , [2] L. Di Vizio and C. Zhang, On q-summation and confluence, Ann. Inst. Fourier Grenoble, , no. 1, [3] T. Dreyfus and A. Eloy, q-borel-laplace summation for q-difference equations with two slopes, J. Difference Equ. Appl., , no.10, [4] T. Dreyfus, Building meromorphic solutions of q-difference equations using a Borel- Laplace summation, Int. Math. Res. Not. IMRN, 2015, Issue 15, [5] G. Gasper and M. Rahman, Basic Hypergeometric Series 2nd ed., Cambridge, [6] K. Ichinobe, The Borel Sum of Divergent Barnes Hypergeometric Series and its Application to a Partial Differential Equation, Publ. RIMS, Kyoto Univ., , [7] L. J. Slater, General transformations of bilateral series, Q. J. Math., , [8] T. Morita, The Stokes phenomenon for the q-difference equation satisfied by the basic hypergeometricseries 3 ϕ 1 a 1,a 2,a 3 ;b 1 ;q,x,rims KôkyûrokuBessatsuB472014, [9] J. P. Ramis, About the growth of entire functions solutions of linear algebraic q-difference equations, Ann. Fac. Sci. Toulouse Math. 6, , no. 1, [10] C. Zhang, Une sommation discrète pour des équations aux q-defférences linéaires et à coefficients analytiques: théorie générale et examples, Differential Equations and Stokes Phenomenon, World Sci. Publ., River Edge, NJ, 2002, [11] C. Zhang, Remarks on some basic hypergeometric series, Theory and Applications of Special Functions, Dev. Math., Vol. 13, Springer, New York, 2005, Shunya Adachi Graduate School of Education Aichi University of Education Kariya JAPAN address: s217m064@auecc.aichi-edu.ac.jp or s.adachi0324@icloud.com 12