A Connection Formula for the q -Confluent Hypergeometric Function
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2013), 050, 13 pages A Connection Formulafor the q -Conf luent Hypergeometric Function Takeshi MORITAGraduate School of Information Science and Technology, Osaka University,1-1 Machikaneyama-machi, Toyonaka, 560-0043, Japan
E-mail: [email protected]
Received October 09, 2012, in final form July 21, 2013; Published online July 26, 2013http://dx.doi.org/10.3842/SIGMA.2013.050
Abstract.
We show a connection formula for the q -confluent hypergeometric functions ϕ ( a, b ; 0; q, x ). Combining our connection formula with Zhang’s connection formula for ϕ ( a, b ; − ; q, x ), we obtain the connection formula for the q -confluent hypergeometric equa-tion in the matrix form. Also we obtain the connection formula of Kummer’s confluenthypergeometric functions by taking the limit q → − of our connection formula. Key words: q -Borel–Laplace transformation; q -difference equation; connection problem; q -confluent hypergeometric function We show a new connection formula for two independent solutions to the q -confluent hypergeo-metric equation(1 − abqx ) u (cid:0) q x (cid:1) − { − ( a + b ) qx } u ( qx ) − qxu ( x ) = 0 . (1)We use notations in accordance with [2]. Assume that q ∈ C ∗ satisfies 0 < | q | < a/b isnot an integer power of q . The basic hypergeometric series r ϕ s is defined by r ϕ s ( a , . . . , a r ; b , . . . , b s ; q, x ) := (cid:88) n ≥ ( a , . . . , a r ; q ) n ( b , . . . , b s ; q ) n ( q ; q ) n (cid:104) ( − n q n ( n − (cid:105) s − r x n , where ( a ; q ) n is the q -shifted factorial( a ; q ) n := (cid:40) , n = 0 , (1 − a )(1 − aq ) · · · (cid:0) − aq n − (cid:1) , n ≥ , ( a ; q ) ∞ = lim n →∞ ( a ; q ) n , and ( a , a , . . . , a m ; q ) ∞ = ( a ; q ) ∞ ( a ; q ) ∞ · · · ( a m ; q ) ∞ . Equation (1) has solutions u ( x ) = ϕ ( a, b ; − ; q, x ) , u ( x ) = ( abx ; q ) ∞ θ ( − qx ) ϕ (cid:16) qa , qb ; 0; q, abx (cid:17) a r X i v : . [ m a t h . C A ] J u l T. Moritaaround the origin and has solutions v ( x ) = x − α ϕ (cid:16) a, aqb ; q, qabx (cid:17) , v ( x ) = x − β ϕ (cid:18) b, bqa ; q, qabx (cid:19) around infinity. Here q α = a and q β = b .The connection formula for a linear q -difference equation of the second order is a linearrelation between u ( x ), u ( x ) and v ( x ), v ( x ): (cid:18) u ( x ) u ( x ) (cid:19) = (cid:18) C ( x ) C ( x ) C ( x ) C ( x ) (cid:19) (cid:18) v ( x ) v ( x ) (cid:19) , where the connection coefficients C jk ( x ) are q -periodic functions.C. Zhang [10] proposed the connection formula for u ( x ), f ( a, b ; λ, q, x ) = ( b ; q ) ∞ (cid:0) ba ; q (cid:1) ∞ θ ( aλ ) θ ( λ ) θ (cid:0) qaxλ (cid:1) θ (cid:0) qxλ (cid:1) ϕ (cid:16) a, aqb ; q, qabx (cid:17) + ( a ; q ) ∞ (cid:0) ab ; q (cid:1) ∞ θ ( bλ ) θ ( λ ) θ (cid:16) qbxλ (cid:17) θ (cid:0) qxλ (cid:1) ϕ (cid:18) b, bqa ; q, qabx (cid:19) , (2)for x ∈ C ∗ \ [ − λ ; q ]. Here f ( a, b ; λ, q, x ) is the q -Borel–Laplace transform of the divergent series ϕ ( a, b ; − ; q, x ), i.e., f ( a, b ; λ, q, x ) := L + q,λ ◦ B + q ϕ ( a, b ; − ; q, x ) . In this paper, we show the following new connection formula for u ( x ): ϕ ( q/a, q/b ; 0; q, abx ) = ( q/a ; q ) ∞ ( b/a ; q ) ∞ ( aqx, /ax ; q ) ∞ ( abx ; q ) ∞ ϕ ( a, aq/b ; q, /abx )+ ( q/b ; q ) ∞ ( a/b ; q ) ∞ ( bqx, /bx ; q ) ∞ ( abx ; q ) ∞ ϕ ( b, bq/a ; q, /abx ) . Since u ( x ) is a divergent series, the q -Stokes phenomenon appears in Zhang’s connectionformula. But our formula gives the exact relation between the convergent series u ( x ) aroundthe origin and the convergent series v ( x ), v ( x ) around infinity.The theta function of Jacobi is given by the series θ q ( x ) := (cid:88) n ∈ Z q n ( n − x n , ∀ x ∈ C ∗ , we denote θ ( x ) shortly. The theta function is written by the product form θ ( x ) = (cid:16) q, − x, − qx ; q (cid:17) ∞ , which is known as Jacobi’s triple product identity. For any k ∈ Z , the theta function alsosatisfies the q -difference equation θ (cid:0) q k x (cid:1) = q − k ( k − x − k θ ( x ) . The theta function satisfies the inversion formula θ ( x ) = xθ (1 /x ). For all fixed λ ∈ C ∗ , wedefine a q -spiral [ λ ; q ] := λq Z = { λq k ; k ∈ Z } . Note that θ ( λq k /x ) = 0 if and only if x ∈ [ − λ ; q ]. Connection Formula for the q -Confluent Hypergeometric Function 3At first, we review the confluent hypergeometric equation (CHGE). In 1813 [3], C.F. Gaussstudied the hypergeometric series F ( α, β ; γ ; z ) = (cid:88) n ≥ ( α ) n ( β ) n ( γ ) n n ! z n , γ (cid:54) = 0 , − , − , . . . , where ( α ) n = α { α + 1 } · · · { α + ( n − } .More generally, the generalized hypergeometric series is given by r F s ( α , . . . , α r ; β , . . . , β s ; z ) = (cid:88) n ≥ ( α ) n · · · ( α r ) n ( β ) n · · · ( β s ) n n ! z n . The hypergeometric function F ( α, β ; γ ; z ) satisfies the second-order differential equation z (1 − z ) d udz + { γ − ( α + β + 1) z } dudz − αβu = 0 . (3)Gauss gave the connection formula for the function F ( α, β ; γ, z ). We put z (cid:55)→ z/β , take thelimit β → ∞ in equation (3), and obtain the confluent hypergeometric equation (CHGE) z d udz + ( γ − z ) dudz − αu = 0 . (4)Solutions of (4) around the origin areˆ u ( z ) = F ( α ; γ ; z )and ˆ u ( z ) = z − γ F ( α − γ + 1 , − γ, z ) . (5)Solutions around infinity are given by the divergent seriesˆ v ( z ) = ( − z ) − α F ( α, α − γ + 1; − ; 1 /z )and ˆ v ( z ) = e z z α − γ F (1 − α, γ − α ; − ; 1 /z ) . The asymptotic expansion of F ( α ; γ ; z ) is given by F ( α ; γ ; z ) ∼ Γ( γ )Γ( γ − α ) ( − z ) − α F ( α, α − γ + 1; − ; − /z )+ Γ( γ )Γ( α ) e z z α − γ F (1 − α, γ − α ; − ; 1 /z ) , (6)where − π/ < arg z < π/
2. Note that the connection formula for the second solution aroundinfinity (5) can be derived from (6). In Section 2 we deal with another degeneration of equa-tion (3) which is slightly different from the standard way.It is known that there exists a q -analogue of F ( α, β ; γ ; z ), which was introduced by E. Heinein 1847 as ϕ ( a, b ; c ; q, x ) := (cid:88) n ≥ ( a ; q ) n ( b ; q ) n ( c ; q ) n ( q ; q ) n x n . T. MoritaWe assume that c is not integer powers of q . The function ϕ ( a, b ; c ; q, x ) satisfies the second-order q -difference equation ( q -HGE) x ( c − abqx ) D q u + (cid:20) − c − q + (1 − a )(1 − b ) − (1 − abq )1 − q x (cid:21) D q u − (1 − a )(1 − b )(1 − q ) u = 0 , (7)where D q is the q -derivative operator defined for fixed q by D q f ( x ) = f ( x ) − f ( qx )(1 − q ) x . By replacing a , b , c by q α , q β , q γ and then letting q → − , equation (7) tends to the hypergeo-metric equation (3). The q -hypergeometric equation (7) can be rewritten as( c − abqx ) u (cid:0) q x (cid:1) − { c + q − ( a + b ) qx } u ( qx ) + q (1 − x ) u ( x ) = 0 . (8)If we set x (cid:55)→ cx and c → ∞ in (8), we obtain the q -confluent hypergeometric equation (1).Equation (1) is considered as a q -analogue of CHGE. Note that the first solution u ( x ) isa divergent series and u ( x ) is a convergent series around the origin. Therefore we should studythe connection formula for u ( x ) and u ( x ) independently. We need different types of q -Borel–Laplace transformations to obtain the connection formula for u ( x ) and u ( x ). This point isessentially different from the differential equation case.We study connection problems for linear q -difference equations with irregular singular points.The irregularity of q -difference equations are studied using the Newton polygons by J.-P. Ramis,J. Sauloy and C. Zhang [6]. For any q -difference operator P = σ nq + a ( z ) σ n − q + · · · + a n ( z ),the Newton polygon is defined as the convex hull of { ( i, j ) ∈ Z | j ≥ v ( a i ) } , provided that v are z -adic valuation in suitable fields. Graphically, the irregularity of q -difference equations and q -difference modules are the height of the Newton polygon (from the bottom to the upper rightend).Connection problems for linear q -difference equations [5] with regular singular points werestudied by G.D. Birkhoff [1]. Linear q -difference equations have formal power series solutions x α (cid:80) n ≥ a n x n around the origin and x β (cid:80) n ≥ b n x − n around infinity for generic exponents. Butfor connection problems for linear q -difference equations, we replace the function x κ with thefunction θ ( x ) /θ ( kx ), where k = q κ , since these functions satisfy the same q -difference equation σ q f ( x ) = q κ f ( x ). Then, the fundamental system of solutions is given by single valued functionswhich have single poles at suitable q -spirals. Therefore, each connection coefficient has theperiods q and e πi . The first example was given by G.N. Watson [7]. But a few examples ofirregular singular cases are known [4, 8, 9]. In this paper, we give a connection formula for the q -confluent type function using the q -Borel–Laplace transformations.In 1910, Watson [7] showed that the connection formula for the series ϕ ( a, b ; c ; q, x ) has thefollowing form ϕ ( a, b ; c ; q, x ) = ( b, c/a ; q ) ∞ ( c, b/a ; q ) ∞ θ ( − ax ) θ ( − x ) ϕ (cid:16) a, aqc ; aqb ; q, cqabx (cid:17) + ( a, c/b ; q ) ∞ ( c, a/b ; q ) ∞ θ ( − bx ) θ ( − x ) ϕ (cid:18) b, bqc ; bqa ; q, cqabx (cid:19) . (9)Note that we can not set a = 0 or b = 0 directly in this formula.In 2002, C. Zhang [10] showed one of the connection formula for the q -CHGE (2). The q -Borel–Laplace transformations were studied by C. Zhang in [10] (see Section 2 for moredetails). When we study connection problems for q -difference equations, this resummation Connection Formula for the q -Confluent Hypergeometric Function 5method becomes a powerful tool. Note that we can find a new parameter λ in the resumma-tion f ( a, b ; λ, q, x ). Here λ is the direction of the summation. This parameter brings us newviewpoints for the study of the q -Stokes phenomenon.It is known that there exist two different types of the q -Borel–Laplace transformations.The q -Borel–Laplace transformations of the first kind are defined in [10] and the q -Borel–Laplace transformations of the second kind are studied in [9]. These q -Borel transformationsare formal inverse transformations of each of the q -Laplace transformations.C. Zhang presented a connection formula for the series ϕ ( a, b ; − ; q, x ) by the q -Borel–Laplacetransformations of the first kind. But the connection formula for the second solution of (1) isnot known. In this paper, we show the second connection formula for q -CHGE with the usingof the q -Borel transformation and the q -Laplace transformation of the second kind. Combiningwith Zhang’s connection formula, we obtain the connection formula in the matrix form (seeTheorem 2). Using Watson’s formula (9) we also give another proof of the new connectionformula in Section 2.5.In Section 3 we consider the limit q → − of our connection formula. If we take the limit q → − , we formally obtain the connection formula for the confluent hypergeometric series F . We review a q -confluent hypergeometric equation in Section 2.1. Then we show a connectionformula for the q -confluent hypergeometric function, which is different from Zhang’s formula. For the confluent hypergeometric equation (3), we take another degeneration. We put z (cid:55)→ zγ and take the limit γ → ∞ . Then we obtain z d udz − { − ( α + β + 1) z } dudz + αβu = 0 . (10)Solutions to (10) around the origin are given by the divergent series˜ u ( z ) = F ( α, β ; − , z ) and ˜ u ( z ) = e z ( − z ) − α − β F (1 − α, − β ; − , z ) . Solutions around infinity are given by the convergent series˜ v ( z ) = ( − z ) α F ( α, α − β, − /z ) and ˜ v ( z ) = ( − z ) β F ( β, β − α, − /z ) . We consider a q -analogue of the confluent hypergeometric equation (8). The second-order q -difference equation x { abqx − (1 − q ) } D q u ( x ) + (cid:26) − (1 − a )(1 − b ) − (1 − abq )1 − q x (cid:27) D q u ( x )+ (1 − a )(1 − b )(1 − q ) u ( x ) = 0 . (11)can be rewritten as(1 − abqx ) u (cid:0) xq (cid:1) − { − ( a + b ) qx } u ( xq ) − qxu ( x ) = 0 , (12)which is called a q -confluent hypergeometric equation. When we take q → − , the limit of (11)is the differential equation (8), provided that a = q α , b = q β . T. Morita q -conf luent hypergeometric equation Consider the connection problem of (12). At first we show local solutions for (12) around x = 0and x = ∞ . Lemma 1.
Equation (12) has solutions u ( x ) = ϕ ( a, b ; − ; q, x ) , (13) u ( x ) = ( abx ; q ) ∞ θ ( − qx ) ϕ (cid:16) qa , qb ; 0; q, abx (cid:17) (14) around the origin and solutions v ( x ) = x − α ϕ (cid:16) a, aqb ; q, qabx (cid:17) , (15) v ( x ) = x − β ϕ (cid:18) b, bqa ; q, qabx (cid:19) , (16) around inf inity, provided that a = q α and b = q β . Proof .
We show a fundamental system of solutions of (12) around x = 0. If set u ( x ) = (cid:80) n ≥ a n x n , a = 1, then we obtain u ( x ) = ϕ ( a, b ; − ; q, x ) . We set E ( x ) = 1 /θ ( − qx ) and f ( x ) = (cid:80) n ≥ a n x n , a = 1 to obtain another solution solution aroundthe origin. We assume that u ( x ) = E ( x ) f ( x ). Note that the function E ( x ) has the followingproperty σ q E ( x ) = − qx E ( x ) , σ q E ( x ) = q x E ( x ) . Therefore, we obtain the equation (cid:2) q x (1 − abqx ) σ q + q { − ( a + b ) qx } σ q − q (cid:3) f ( x ) = 0 . (17)Since the infinite product ( abx ; q ) ∞ satisfies the following q -difference relation σ q [( abx ; q ) ∞ ] = 11 − abx ( abx ; q ) ∞ , we obtain the second solution. Therefore, solutions of equation (12) around the origin aregiven by u ( x ) = ϕ ( a, b ; − ; q, x ) , u ( x ) = ( abx ; q ) ∞ θ ( − qx ) ϕ (cid:16) qa , qb ; 0; q, abx (cid:17) . Around x = ∞ , we can easily determine local solutions by setting v ( x ) = θ ( aµx ) θ ( µx ) (cid:88) n ≥ a n x − n , a = 1 , for any fixed µ ∈ C ∗ and x ∈ C ∗ \ [ − µ ; q ]. (cid:4) Here u ( x ) is a divergent series and u ( x ), v ( x ) and v ( x ) are convergent series [2]. Therefore,the q -Stokes phenomenon occurs for u ( x ). Connection Formula for the q -Confluent Hypergeometric Function 7 Definition 1.
For any f ( x ) = (cid:80) n ≥ a n x n , the q -Borel transformation B + q is (cid:0) B + q f (cid:1) ( ξ ) = ϕ ( ξ ) := (cid:88) n ≥ a n q n ( n − ξ n , and the q -Laplace transformation L + q,λ is (cid:0) L + q,λ ϕ (cid:1) ( x ) := (cid:88) n ∈ Z ϕ ( q n λ ) θ (cid:16) q n λx (cid:17) . C. Zhang determined a resummation of (13) by the q -Borel–Laplace transformations of thefirst kind as follows f ( a, b ; λ, q, x ) := L + q,λ ◦ B + q ϕ ( a, b ; − ; q, x ) . He also presented a connection formula (2) for f ( a, b ; λ, q, x ).But the connection formula between (14) and (15), (16) is not known. In the next section,we show the second connection formula by means of the q -Borel–Laplace transformations of thesecond kind. We define the q -Borel transformation and the q -Laplace transformation of the second kind.These transformations are introduced by C. Zhang to obtain the solution of equation (17). Definition 2.
For f ( x ) = (cid:80) n ≥ a n x n , the q -Borel transformation is defined by g ( ξ ) = (cid:0) B − q f (cid:1) ( ξ ) := (cid:88) n ≥ a n q − n ( n − ξ n , and the q -Laplace transformation is (cid:0) L − q g (cid:1) ( x ) := 12 πi (cid:90) | ξ | = r g ( ξ ) θ q (cid:18) xξ (cid:19) dξξ . Here r > q -Borel transformation is considered asa formal inverse of the q -Laplace transformation. Lemma 2 ([9]) . We assume that the function f can be q -Borel transformed to the analyticfunction g ( ξ ) around ξ = 0 . Then, we have L − q ◦ B − q f = f. Proof .
We can prove this lemma calculating residues of the q -Laplace transformation aroundthe origin. (cid:4) The q -Borel transformation satisfies the following operational relation. Lemma 3.
For any l, m ∈ Z ≥ , B − q (cid:0) x m σ lq (cid:1) = q − m ( m − ξ m σ l − mq B − q . T. MoritaWe apply the q -Borel transformation to equation (17) and use Lemma 3. We use the no-tation g ( ξ ) as the q -Borel transform of u ( x ). We check out that g ( ξ ) satisfies the first-order q -difference equation g ( qξ ) = (1 + aqξ )(1 + bqξ )(1 + q ξ ) g ( ξ ) . Since g (0) = a = 1, we have the infinite product of g ( ξ ) as follows g ( ξ ) = ( − q ξ ; q ) ∞ ( − qaξ ; q ) ∞ ( − qbξ ; q ) ∞ . Note that g ( ξ ) has single poles at (cid:26) ξ ∈ C ∗ ; ξ = − aq k +1 , − bq k +1 , k ∈ Z ≥ (cid:27) . We set r := max (cid:26) | aq | , | bq | (cid:27) and choose the radius r > < r < r . By Cauchy’s residue theorem, the q -Laplacetransform of g ( ξ ) is f ( x ) = 12 πi (cid:90) | ξ | = r g ( ξ ) θ (cid:18) xξ (cid:19) dξξ = − (cid:88) k ≥ Res (cid:26) g ( ξ ) θ (cid:18) xξ (cid:19) ξ ; ξ = − aq k +1 (cid:27) − (cid:88) k ≥ Res (cid:26) g ( ξ ) θ (cid:18) xξ (cid:19) ξ ; ξ = − bq k +1 (cid:27) , where 0 < r < r . Since there exists a positive constant C N (for any integer N ) s.t., | g ( ξ ) | ≤ C N ξ − N . The following lemma plays a key role to calculate the residue.
Lemma 4.
For any k ∈ N , λ ∈ C ∗ , we have:
1) Res (cid:26) ξ/λ ; q ) ∞ ξ : ξ = λq − k (cid:27) = ( − k +1 q k ( k +1)2 ( q ; q ) k ( q ; q ) ∞ ,
2) 1( λq − k ; q ) ∞ = ( − λ ) − k q k ( k +1)2 ( λ ; q ) ∞ ( q/λ ; q ) k , λ (cid:54)∈ q Z . Summing up all residues, we obtain f ( x ) as follows f ( x ) = (cid:0) qa ; q (cid:1) ∞ (cid:0) ba , q ; q (cid:1) ∞ θ ( − aqx ) θ ( − qx ) ϕ (cid:16) a, aqb ; q, qabx (cid:17) + (cid:0) qb ; q (cid:1) ∞ (cid:0) ab , q ; q (cid:1) ∞ θ ( − bqx ) θ ( − qx ) ϕ (cid:18) b, bqa ; q, qabx (cid:19) . Therefore, we obtain the following theorem.
Theorem 1.
For any x (cid:54)∈ [1; q ] , we have u ( x ) = ( abx ; q ) ∞ θ ( − qx ) ϕ (cid:16) qa , qb ; 0; q, abx (cid:17) = (cid:0) qa ; q (cid:1) ∞ (cid:0) ba , q ; q (cid:1) ∞ θ ( − aqx ) θ ( − qx ) ϕ (cid:16) a, aqb ; q, qabx (cid:17) + (cid:0) qb ; q (cid:1) ∞ (cid:0) ab , q ; q (cid:1) ∞ θ ( − bqx ) θ ( − qx ) ϕ (cid:18) b, bqa ; q, qabx (cid:19) . (18) Connection Formula for the q -Confluent Hypergeometric Function 9 Combining Zhang’s connection formula and Theorem 1, we give the connection matrix for equa-tion (1). At first, we define a new fundamental system of solutions around infinity. For any λ , µ ∈ C ∗ , x ∈ C ∗ \ [ − µ ; q ], S µ ( a, b ; q, x ) is S µ ( a, b ; q, x ) := θ ( aµx ) θ ( µx ) ϕ (cid:16) a, aqb ; q, qabx (cid:17) . The function θ ( aµx ) /θ ( µx ) satisfies the following q -difference equation u ( qx ) = 1 a u ( x ) , which is also satisfied by the function u ( x ) = x − α , a = q α . Note that the pair ( S µ ( a, b ; q, x ) , S µ ( b, a ; q, x )) gives a fundamental system of solutions of equation (1) if a/b (cid:54)∈ q Z . We define q -elliptic functions C λµ ( a, b ; q, x ) and C µ ( a, b ; q, x ). Definition 3.
For any λ, µ ∈ C ∗ we set functions C λµ ( a, b ; q, x ) and C µ ( a, b ; q, x ) as follows C λµ ( a, b ; q, x ) := ( b ; q ) ∞ (cid:0) ba ; q (cid:1) ∞ θ ( aλ ) θ ( λ ) θ (cid:0) qaxλ (cid:1) θ (cid:0) qxλ (cid:1) θ ( µx ) θ ( aµx ) ,C µ ( a, b ; q, x ) := (cid:0) qa ; q (cid:1) ∞ (cid:0) ba , q ; q ∞ (cid:1) θ ( − aqx ) θ ( − qx ) θ ( µx ) θ ( aµx ) . Then C λµ ( a, b ; q, x ) and C µ ( a, b ; q, x ) are single valued as functions of x . They satisfy thefollowing relation C λµ ( a, b ; q, e πi x ) = C λµ ( a, b ; q, x ) , C λµ ( a, b ; q, qx ) = C λµ ( a, b ; q, x )and C µ (cid:0) a, b ; q, e πi x (cid:1) = C µ ( a, b ; q, x ) , C µ ( a, b ; q, qx ) = C µ ( a, b ; q, x ) . i.e. C λµ ( a, b ; q, x ) and C µ ( a, b ; q, x ) are q -elliptic functions. We set f ( a, b ; q, x ) := u ( x ) = ( abx ; q ) ∞ θ ( − qx ) ϕ (cid:16) qa , qb ; 0; q, abx (cid:17) . Thus, we obtain the connection formula in the matrix form.
Theorem 2.
For any λ, µ ∈ C ∗ , x ∈ C ∗ \ [1; q ] ∪ [ − µ/a ; q ] ∪ [ − µ/b ; q ] ∪ [ − λ ; q ] , we have (cid:18) f ( a, b ; λ, q, x ) f ( a, b ; q, x ) (cid:19) = (cid:18) C λµ ( a, b ; q, x ) C λµ ( b, a ; q, x ) C µ ( a, b ; q, x ) C µ ( b, a ; q, x ) (cid:19) (cid:18) S µ ( a, b ; q, x ) S µ ( b, a ; q, x ) (cid:19) . In this section, we give another proof of Theorem 1. Watson’s formula (9) is a connection formulafor the basic hypergeometric functions ϕ ( a, b ; c ; q, x ). We derive the connection formula( abx ; q ) ∞ θ ( − qx ) ϕ (cid:16) qa , qb ; 0; q, abx (cid:17) = ( q/a ; q ) ∞ ( b/a, q ; q ) ∞ θ ( − aqx ) θ ( − qx ) ϕ (cid:16) a, aqb ; q, qabx (cid:17) + ( q/b ; q ) ∞ ( a/b, q ; q ) ∞ θ ( − bqx ) θ ( − qx ) ϕ (cid:18) b, bqa ; q, qabx (cid:19) from Watson’s formula (9). By take the limit c → Proposition 1 ([10]) . For any x ∈ C ∗ \ q Z , we have ϕ ( a, b ; 0; q, x ) = ( b ; q ) ∞ ( b/a ; q ) ∞ θ ( − ax ) θ ( − x ) ϕ (cid:18) a ; aqb ; q, q bx (cid:19) + ( a ; q ) ∞ ( a/b ; q ) ∞ θ ( − bx ) θ ( − x ) ϕ (cid:18) b ; bqa ; q, q ax (cid:19) . (19)In (19), we put a (cid:55)→ q/a , b (cid:55)→ q/b and x (cid:55)→ abx , we obtain the relation as follows Corollary 1.
For any x ∈ C ∗ \ [ ab ; q ] , we have ϕ (cid:16) qa , qb ; 0; q, abx (cid:17) = ( q/b ; q ) ∞ ( a/b ; q ) ∞ θ ( − bqx ) θ ( − abx ) ϕ (cid:18) qa ; bqa ; q, qax (cid:19) + ( q/a ; q ) ∞ ( b/a ; q ) ∞ θ ( − aqx ) θ ( − abx ) ϕ (cid:16) qb ; aqb ; q, qbx (cid:17) . (20)The function ϕ ( a ; c ; q, x ) is related to ϕ ( a (cid:48) , c (cid:48) ; q, x ). Consider the relation between thefunction ϕ and the function ϕ . Proposition 2.
For any x ∈ C ∗ , we have ϕ (cid:18) a ; c ; q, c xa (cid:19) = ( x ; q ) ∞ ϕ (cid:18) c a , c ; q, x (cid:19) , (21) provided that c /a (cid:54)∈ q Z . Proof .
The function ϕ ( a ; c ; q, c x/a ) satisfies the equation (cid:20) ( c − c qx ) σ q − (cid:26) ( c + q ) − qc a x (cid:27) σ q + q (cid:21) v ( x ) = 0 . We set v ( x ) = ( x ; q ) ∞ ˜ v ( x ), where ˜ v ( x ) := (cid:80) n ≥ ˜ v n x n and ˜ v := 1. Note that the function ( x ; q ) ∞ satisfies the first-order q -difference equation σ q f ( x ) = 11 − x f ( x ) . Then, we obtain the equation (cid:20) σ q − (cid:26)(cid:18) qc (cid:19) − qxa (cid:27) σ q + qc (1 − x ) (cid:21) ˜ v ( x ) = 0 . (22)Equation (22) has the solution˜ v ( x ) = ϕ (cid:18) c a , c ; q, x (cid:19) . Therefore, we obtain the conclusion. (cid:4)
Corollary 2. In (21) , we put a (cid:55)→ q/a , c (cid:55)→ bq/a and x (cid:55)→ q/abx . Then we obtain ϕ (cid:18) qa ; bqa ; q, qax (cid:19) = (cid:16) qabx ; q (cid:17) ∞ ϕ (cid:18) b, bqa ; q, qabx (cid:19) . (23) We also put a (cid:55)→ q/b , c (cid:55)→ aq/b and x (cid:55)→ q/abx . Then we obtain ϕ (cid:16) qb ; aqb ; q, qbx (cid:17) = (cid:16) qabx ; q (cid:17) ∞ ϕ (cid:16) a, aqb ; q, qabx (cid:17) . (24) Connection Formula for the q -Confluent Hypergeometric Function 11By relations (20), (23) and (24),( abx ; q ) ∞ θ ( − qx ) ϕ (cid:16) qa , qb ; 0; q, abx (cid:17) = ( q/b ; q ) ∞ ( a/b, q ; q ) ∞ θ ( − bqx ) θ ( − abx ) (cid:0) q, abx, qabx ; q (cid:1) ∞ θ ( − qx ) ϕ (cid:18) b, bqa ; q, qabx (cid:19) + ( q/a ; q ) ∞ ( b/a, q ; q ) ∞ θ ( − aqx ) θ ( − abx ) (cid:0) q, abx, qabx ; q (cid:1) ∞ θ ( − qx ) ϕ (cid:16) a, aqb ; q, qabx (cid:17) = ( q/b ; q ) ∞ ( a/b, q ; q ) ∞ θ ( − bqx ) θ ( − qx ) ϕ (cid:18) b, bqa ; q, qabx (cid:19) + ( q/a ; q ) ∞ ( b/a, q ; q ) ∞ θ ( − aqx ) θ ( − qx ) ϕ (cid:16) a, aqb ; q, qabx (cid:17) . Therefore, we obtain the formula (18). q → − of the connection formula In this section we show the limit q → − of our connection formula. In [10], C. Zhang proposedthe following limit. Theorem 3 ([10]) . For any α, β ∈ C ∗ ( α − β (cid:54)∈ Z ) and z in any compact domain of C ∗ \ [ −∞ , ,we have lim q → − f (cid:18) q α , q β ; λ, q, z (1 − q ) (cid:19) = Γ( β − α )Γ( β ) z − α F (cid:18) α, α − β + 1; 1 z (cid:19) + Γ( α − β )Γ( α ) z − β F (cid:18) β ; β − α + 1; 1 z (cid:19) . Our limit of the connection formula in Theorem 1 differs from the theorem above. ByTheorem 1, we have u ( x ) = C µ ( a, b ; q, x ) S µ ( a, b ; q, x ) + C µ ( b, a ; q, x ) S µ ( b, a ; q, x ) (25)for any ( x, q ) ∈ C ∗ × (0 , q → − of the left-hand side of (25) is formally given by e /z ( − z ) − α − β F (1 − α, − β ; − , z ), provided that a = q α , b = q β and x = z/ (1 − q ). On the other hand, convergentseries F ( α ; α +1 − β ; 1 /z ) and F ( β ; β +1 − α ; 1 /z ) appear in the limit q → − of the right-handside of (25). The aim of this section is to prove the following theorem. Theorem 4.
The limit q → − of the new connection formula formally gives the followingasymptotic formula e /z ( − z ) − α − β F (1 − α, − β ; − , z ) = Γ( β − α )Γ(1 − α ) ( − z ) − α F (cid:18) α ; α + 1 − β ; 1 z (cid:19) + Γ( α − β )Γ(1 − β ) ( − z ) − β F (cid:18) β ; β + 1 − α ; 1 z (cid:19) . In [8], Zhang has shown a limit of theta functions, taking the principal value of the logarithmon C ∗ \ ( −∞ , Proposition 3.
For any γ ∈ C ∗ , we have lim q → − θ (cid:16) q γ u − q (cid:17) θ (cid:16) u − q (cid:17) (1 − q ) − γ = u − γ converges uniformly on compact subset of C \ ( −∞ , . We also remind the formulas for the q -gamma function Γ q ( · ) and the q -exponential func-tion E q ( · ). The q -gamma function is defined byΓ q ( x ) := ( q ; q ) ∞ ( q x ; q ) ∞ (1 − q ) − x , < q < . This function satisfies lim q → − Γ q ( x ) = Γ( x ) [2]. The q -exponential function E q ( z ) = (cid:88) n ≥ q n ( n − / ( q ; q ) n z n = ( − z ; q ) ∞ satisfies the limitlim q → − E q ( z (1 − q )) = e z . We set a = q α , b = q β and x = z/ (1 − q ) in Theorem 1. We introduce the constant w ( α, β ; q ) := ( q ; q ) ∞ (1 − q ) − α − β . Consider the limit when q → − of each side of the identity of Theorem 1. The limit of the lefthand side of (18) is given by the following lemma. Lemma 5.
For any α, β ∈ C ∗ , α − β (cid:54)∈ Z , we have lim q → − w ( α, β ; q ) (cid:16) q α + β z − q ; q (cid:17) ∞ θ (cid:16) − qz − q (cid:17) ϕ (cid:18) q − α , q − β ; 0; q, q α + β z − q (cid:19) = ( − z ) − α − β e z F (1 − α, − β ; − , z ) . Proof .
Exploiting the fact w ( α, β ; q ) (cid:16) q α + β z − q ; q (cid:17) ∞ θ (cid:16) − qz − q (cid:17) ϕ (cid:18) q − α , q − β ; 0; q, q α + β z − q (cid:19) = θ (cid:16) q α + β (cid:16) − z − q (cid:17)(cid:17) θ (cid:16) − z − q (cid:17) (1 − q ) − α − β θ (cid:16) − z − q (cid:17) θ (cid:16) q (cid:16) − z − q (cid:17)(cid:17) (1 − q ) × E q (cid:16) − (1 − q ) q α + β − z (cid:17) ϕ (cid:18) q − α , q − β ; 0; q, q α + β z − q (cid:19) , we obtain the conclusion. (cid:4) Consider the right-hand side of (18). Connection Formula for the q -Confluent Hypergeometric Function 13 Lemma 6.
For any α, β ∈ C ∗ , ( α − β (cid:54)∈ Z ) , we have lim q → − w ( α, β, q ) f (cid:18) q α , q β ; q, z (1 − q ) (cid:19) = Γ( β − α )Γ(1 − α ) ( − z ) − α F (cid:18) α ; α + 1 − β ; 1 z (cid:19) + Γ( α − β )Γ(1 − β ) ( − z ) − β F (cid:18) β ; β + 1 − α ; 1 z (cid:19) . Proof .
Noting that w ( α, β ; q ) (cid:0) q − α ; q (cid:1) ∞ ( q β − α , q ; q ) ∞ θ (cid:16) − q α +1 z − q (cid:17) θ (cid:16) − qz − q (cid:17) ϕ (cid:18) q α , q α +1 − β ; q, q − α − β (1 − q ) z (cid:19) = (cid:26) ( q − α ; q ) ∞ ( q ; q ) ∞ (1 − q ) α (cid:27) (cid:26) ( q ; q ) ∞ ( q β − α ; q ) ∞ (1 − q ) − (1 − β + α ) (cid:27) × θ (cid:16) q α +1 (cid:16) − z − q (cid:17)(cid:17) θ (cid:16) − z − q (cid:17) (1 − q ) − α − θ (cid:16) − z − q (cid:17) θ (cid:16) q (cid:16) − z − q (cid:17)(cid:17) (1 − q ) × ϕ (cid:18) q α , q α +1 − β ; q, q − α − β (1 − q ) z (cid:19) , we prove the lemma. (cid:4) Finally, we obtain the proof of Theorem 4.
Acknowledgements
The author would like to give heartfelt thanks to Professor Yousuke Ohyama who providedcarefully considered feedback and many valuable comments. The author also would like tothank the anonymous referees for their helpful comments.
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