Remarks on Some Basic Hypergeometric Series

Transcription

1 Remarks on Some Basic Hypergeometric Series Changgui Zhang Many results in Mathematical Analysis seem to come from some obvious computations. For a few years, we have been interested in the analytic theory of linear q-difference equations. One of the problems we are working on is the analytical classification of q-difference equations. Recall that this problem was already considered by G.D. Birkhoff and some of his students ([2], [3]). An important stage of these works consists to be able to get transcendental analytical invariants from the divergent power series solutions; that is, to be able to define a good concept of Stokes multiplier for divergent q-series! Very recently, we noted ([11], [7]) that this problem can be agreeably solved by a new summation theory of divergent power series with the help of a theta function of Jacobi and some basic integral calculus. The purpose of the present article is to explain how much obvious this mechanism of summation may be if one practises some elementary calculations on q-series. It would be a very interesting question to understand ([4]) whether Ramanujan s mysterious formulas are related to this transcendental invariant analysis... The article contains four sections. In the first section, we explain how to use the θ function for giving a q-integral representation which remains valid for the sum function of every convergent power series. It is this integral representation which leads us to a new process of summation of divergent series. Some identities then follow on the convergent power series, in the spirit of a Stokes analysis: the convergence takes place only in spite of the Stokes phenomenon! It is well known that every basic hypergeometric series 2 ϕ 0 is formal it of a family of 2 ϕ 1 s. In [11], we have applied the new summation method to this divergent power series 2 ϕ 0. In the third section of our present article, we prove that each sum of 2 ϕ 0 is exactly the it of an associated family of 2 ϕ 1 while the parameter tends to infinity following a q-spiral. In the last section, we give a remark about the Euler s Γ function. The best known q-analog of Γ is certainly the Jackson s Γ q ([1]), which satisfies a first order q-difference equation deduced from the fundamental equation of Γ. This q-difference equation has a formal Laurent series solution that is divergent everywhere in C ([10]). Using a summation formula of Ramanujan for 1 ψ 1 and the above-mentioned summation method, one gets a new q-analog of Γ which is meromorphic on C. Some basic notations. In the following, q denotes a real number in the open interval (0, 1) and some notations of the book [5] will be used. For example, if Laboratoire AGAT (UMR CNRS 8524), UFR Math., Université des Sciences et Technologies de Lille, Cité Scientifique, Villeneuve d Ascq cedex, France. zhang@agat.univ-lille1.fr 1

2 a C, we set (a; q) 0 = 1, for a 1,..., a l C, we set: n 1 (a; q) n = (1 aq m ), n N {+ } ; m=0 (a 1,..., a l ; q) n = (a 1 ; q) n... (a l ; q) n, n N {+ }. To each fixed C, we associate its so-called q-spiral [; q] by setting q Z = [; q] = {q n : n Z}. If, µ C, the following conditions are equivalent: [; q] [µ; q] [; q] = [µ; q] /µ q Z. 1 How to get the sum of a power series by means of θ Let us denote by θ q (x) or more shortly θ(x) the theta function of Jacobi, given by the following series: θ(x) = n Z q n(n 1)/2 x n, x C. Jacobi s triple product formula says that: θ(x) = (q, x, q/x; q) ; from this, it follows that θ(x) = 0 if and only if x [ 1; q]. Recall also that θ verifies the fundamental equation θ(x) = xθ(qx) or, more generally, for any n Z θ(q n x) = q n(n 1)/2 x n θ(x) (1) Consider any given power series f = n 0 a nx n with complex coefficients and let be an arbitrary nonzero complex number. Suppose the radius R of convergence of f is > 0. It is obvious that the product f(x)θ(/x) defines an analytical function in the truncated disc 0 < x < R. From direct computations, one obtains the following identity: f(x)θ( x ) = n Z ϕ f (q n )q n(n 1)/2 ( x )n, (2) where ϕ f denotes the entire function, depending upon f, defined as follows: ϕ f (ξ) def = n 0 a n q n(n 1)/2 ξ n. (3) It is useful to note that for any B > 1/R, there exists C > 0 such that ξ C, ϕ f (ξ) < Cθ(B ξ ). 2

3 According to [6], the function ϕ f is said to have a q-exponential growth of order (at most) one at infinity. Let C{x} be the ring of all power series that converge near x = 0 and E q;1 the set of all entire functions having at most a q-exponential growth of order one at infinity. Proposition 1.1 The map f ϕ f given in (3) establishes a bijection between C{x} and E q;1. More precisely, if ϕ = ϕ f, f C{x} and if R > 0 is the radius of convergence of f, then the following assertions hold. 1. For any 0 < r < R, let C 0,r + be the counterclockwise-oriented circle centered at the origin and of radius r, then ϕ(ξ) = 1 f(x)θ( ξ 2πi x )dx x, ξ C. (4) C + 0,r 2. For any C, the following q-integral 1 ϕ(ξ) d q ξ 1 q 0 θ( ξ x ) ξ def = ϕ(q n ) θ( qn n Z x ) (5) defines an analytical function which is equal to the sum of the convergent power series f in the open disc 0 < x < R minus the points of the set ( q Z ) (i.e. the q-spiral [ ; q]). Accordingly, one can obtain the sum of a convergent power series by the following process: f ϕ = ϕ f 1 ϕ(ξ) d q ξ 1 q 0 θ( ξ x ) ξ (= Sf). (6) Proof - Applying Cauchy s formula to ϕ f gives the growth of its coefficients, from which we deduce that the map f ϕ f is surjective (see [6]). It is obvious that this map is also injective. Note that the q-integral representation of 2. is a reformulation of the identity (2), while the formula (4) follows directly from Cauchy. According to [11] and [8], we shall denote by ˆB q;1 f the power series ϕ f of (3) and by L [;q] q;1 ϕ the q-integral of (5). It is important to remark that the convergence of L [;q] q;1 ϕ depends only on the asymptotic behaviour of the function ϕ(ξ) as ξ goes to along the q-spiral [; q]. So, let H [;q] q;1 be the set of analytic function germs at the origin that can be analytically continued to a function having a q-exponential growth of order one at infinity in a neighborhood of [; q]. Here we call neighborhood of [; q] any domain V C such that there exists a neighborhood U of in C for which the inclusion {ξq n : ξ U, n Z} V holds. It is essential to notice that E q;1 H [;q] q;1 for every C, the inclusion being strict. Therefore, the process (6) allows us to sum not only the convergent power series, but also any power series ˆf that can be transformed by ˆf ˆB q;1 ˆf 3

4 1 to be an element of H [;q] q;1. By definition, these power series ˆf are called [; q]-summable, of sum L [;q q;1 ˆB q;1 ˆf. In [11], it is shown that the summation process ˆf L [;q] q;1 ˆB q;1 ˆf can be applied to every formal power series solution of any q-difference equation if this equation is, in some sense, generically singular. For example, all basic hypergeometric series 2 ϕ 0 (a, b; ; q, x) are summable by this method. For more details, see [11] and [8]. 2 Identities characterizing the convergence of a power series Let ϕ = ϕ f, f C{x} (i.e. f is a convergent power series) and suppose R is the radius of convergence of f. Since θ( 1) = 0, putting = ξ and x = ξ in (2) gives the following formula: ( 1) n q n(n 1)/2 ϕ(q n ξ) = 0, (7) n Z which is valid for any ξ C such that ξ < R. The equality (7) can be viewed as an identity characterizing the convergence of the power series f such that ϕ = ϕ f. Indeed, let ϕ be any analytical function known in a domain U of the form U = {ξ C : ξ < 1} {ξ C : α < arg x < α}, where α (0, π). Suppose that ϕ has a q-exponential growth of order at most one at infinity. For each x U such that α < arg x < α and x < R (for a suitable fixed real R > 0), we write ϕ(ξ) = n Z( 1) n q n(n 1)/2 ϕ(q n ξ). Theorem 2.1 Let V α;r = {ξ C : α < arg x < α, ξ < R}. The function ϕ is identically equal to zero on the sector V α;r if and only if there exists a convergent power series f (i.e. f C{x}) such that ϕ = ϕ f. Proof - The if part has been explained at the beginning of this section. We shall prove the only if part. For each δ ( α, α), we note D δ R = {x C : x < R, δ π < arg x < δ+π} and we set, if x D δ R : f δ (x) = 1 e iδ ϕ(ξ) dξ ln(1/q) 0 θ( ξ x ) ξ. (8) Since ϕ is holomorphic in U, the functions f δ can be glued into an analytic function on the sector V = {x C : x < R, α π < arg x < α + π} of the Riemann surface of the logarithm function. Let f be this function just constructed on V ; by the theorem of residues, we get: f(xe πi ) f(xe πi ) = 2πi ln(1/q) ϕ(ξ) 1 Here one uses the notation ˆf instead of f, because the series under consideration is not necessarily convergent: one will have to distinguish a power series from its sum(s). 4

5 for all x V α,r. By assumption, ϕ(ξ) = 0 on V α;r ; hence, it follows that f can be identified to an analytical function in the truncated disc 0 < x < R. We write again f for the latter function. By means of direct estimations for (8), one can check that f is bounded in a neighborhood of zero. Therefore, by Riemann s Removable Singularities Theorem the function f is holomorphic at the origin, i.e. f is the sum function of a convergent power series in the disc x < R. It only remains to verify that the Taylor expansion of ϕ at zero coincides with the power series ϕ f. To do this, one can use the following formula: for more details, see [9]. 1 e iδ ξ n dξ ln(1/q) 0 θ( ξ x ) ξ = q n(n 1)/2 ξ n ; Now let s go back again to the formula (2) and let be given C, µ C ; one gets: θ( µ x ) n Z ϕ(q n )q n(n 1)/2 ( x )n = θ( x ) n Z ϕ(µq n )q n(n 1)/2 ( µ x )n. (9) Recall that for any analytic function g given in an open disc 0 < x < r, if n Z a nx n is its Laurent series expansion, then the formula (2) can be extended in the following way: θ( x )g(x) = m Z a n q n(n 1)/2 (q m ) n q m(m 1)/2 ( x )m. (10) n Z At the same time, the equality (9) takes the following form: ϕ(q n )q n2 ( µq m )n q m(m 1)/2 ( µ x )m = ϕ(µq n )q n2 ( µ q m )n q m(m 1)/2 ( x )m ; m Z n Z m Z n Z this implies: m Z, µ m n Z ϕ(q n )( µ )n q n(n m) = m n Z ϕ(µq n )( µ )n q n(n m). So, C, µ C, ɛ {0, 1/2}: ϕ(q n )( µ )n q n2 = ϕ(µq n )( µ )n q n2. (11) n ɛ+z n ɛ+z The identity (11) is also a formula that characterizes the convergence of the power series f such that ϕ = ϕ f (= ˆB q;1 f). Indeed, consider any function ϕ H [;q] q;1 ; it may be noticed that ϕ H[µ;q] q;1 for all µ C close enough to. The following result is essentially related to the PRINCIPLE that the [; q]-summation is a totally discontinuous mapping with respect to unless the power series to sum has in fact a radius of convergence > 0! 5

6 Theorem 2.2 Let C and consider a function ϕ H [;q] q;1. We have ϕ E q;1 if and only if there exists µ C such that [; q] [µ; q], ϕ H [µ;q] q;1 and that (11) holds for ɛ = 0 and 1/2. Proof - The identity (11) holds if, and only if, according to (9-10), we have the following: L [;q] q;1 ϕ = L[µ;q] q;1 ϕ. Remember that each integral L [;q] q;1 ϕ defines an analytic function in the truncated disc 0 < x < R minus all points of the q-spiral [ ; q]. Hence, neither ϕ nor L[µ;q] q;1 ϕ has singularity in the disc 0 < x < R. The proof may be completed by the Removable Singularities Theorem and the fact that L [;q] q;1 ϕ is asymptotic to the power series ˆf such that ϕ = ˆB q;1 ˆf; see [11] and [8]. L [;q] q;1 It is obvious that the formulas (7), (11) can be extended to a very larger class of functions that may possess a singularity at zero; cf. (10). On the other hand, if we only restricted to the class of rational functions ϕ = P/Q (P, Q C[ξ]), the equalities (7) and (11) would give criteria on the divisibility of P by Q: what unexpected criteria (how to make them effective?)! 3 Confluence of 2 ϕ 1 to 2 ϕ 0 along a q-spiral For any a, b, c C such that c q N, we write 2 ϕ 1 (a, b; c; q; x), as in the book [5], the following Heine s series: 2ϕ 1 (a, b; c; q; x) = n 0 (a; q) n (b; q) n (q; q) n (c; q) n x n. If ab 0 and a/b / q Z, the sum of the power series 2 ϕ 1 (a, b; c; q, x) can be analytically continued in the cut plane C \ [1, + ) by the following formula, due to G.N. Watson: 2ϕ 1 (a, b; c; q, x) = (b, c/a; q) (c, b/a; q) + (a, c/b; q) (c, a/b; q) θ( ax) cq θ( x) 2 ϕ 1 (a, aq/c; aq/b; q, abx ) + θ( bx) cq θ( x) 2 ϕ 1 (b, bq/c; bq/a; q, abx ). (12) Note that if c, the series 2 ϕ 1 (a, b; c; q, ct) converges termwise to 2 ϕ 0 (a, b; ; q, t), where 2ϕ 0 (a, b; ; q, t) = n 0 (a, b; q) n (q; q) n q n(n 1)/2 ( t) n. (13) Unless a or b q N, the last power series is divergent for all t 0 and, according to Theorem [11], it is [; q]-summable for all C \( q Z ). More precisely, let 2 f 0 (a, b;, q, t) be the sum of 2 ϕ 0 (a, b; ; q, t) corresponding to the path [; q]. By Theorem [11], one has the following formula, which is similar to 6

7 the above-mentioned one (12): 2f 0 (a, b;, q, t) = (b; q) (b/a; q) θ(a) θ() + (a; q) (a/b; q) θ(qat/) θ(qt/) 2 ϕ 1 (a, 0; aq b ; q, q abt ) + θ(b) θ(qbt/) θ() θ(qt/) 2 ϕ 1 (b, 0; bq a ; q, q abt ). (14) Theorem 3.1 Let C \ ( q Z ). For all t C \ ( q Z ), one has: n N n 2ϕ 1 (a, b; q n ; q, q n t) = 2f 0 (a, b;, q, t). (15) Remark that this Theorem has a fast proof: one verifies that the right side of (12) converges, as c = q n, n N, x = ct and n +, to the right side of (14); to do this, the following lemma is helpful. Lemma 3.2 Let, a C such that [ 1; q]. One has the following: ( aq n ; q) n N ( q n a n = θ(a) ; q) θ(). (16) n Therefore, if α, β, γ and δ are four complex numbers such that αβ = δγ 0, δ [ 1; q], γ [ 1; q], then the following it holds: n N n ( αq n, βq n ) ( γq n, δq n = θ(α)θ(β) ; q) θ(γ)θ(δ). Proof - It suffices to notice that, for any n N one has: ( q n ; q) = ( ; q) ( 1 ; q) n n q n(n+1)/2. Now, we shall give a direct proof of Theorem 3.1: it is helpful to understand in which way the formula (4) goes to its it form after the confluence along a q-spiral. where Proof of Theorem Let n N; thanks to the formula (2), one has: θ( t ) 2ϕ 1 (a, b; q n ; q, q n t) = ϕ fn (q m )q m(m 1)/2 ( t )m, m Z ϕ fn (q m ) = 2 ϕ 2 (a, b; q n, 0; q, q n+m ) 1 = 2ϕ 1 (a, b; q n ; q, x)θ( q n+m ) dx 2πi C + x x. (17) 0;r Next, one expands the function 2 ϕ 1 (a, b; q n ; q, x) at infinity by means of the formula (12). Hence, the integral of (17) gives the following: ϕ fn (q m ) = (b, q n /(a); q) ( q n /, b/a; q) + (a, q n /(b); q) ( q n /, a/b; q) a 1 m+n 2ϕ 2 (a, aq n+1 ; aq b 7 b 1 m+n 2ϕ 2 (b, bq n+1 ; bq a, 0; q, q2 m ) + b, 0; q, q2 m a ).

8 which, together with the formula (16), leads to the conclusion of Theorem q-analogs of Γ and summation of a basic bilateral series The Jackson s q-gamma function satisfies the functional equation Γ q (z) = (q; q) (q z ; q) (1 q) 1 z G(z + 1) = qz 1 G(z). (18) q 1 Letting x = q z, this equation leads us to the following q-difference equation: (q 1)y(qx) = (x 1)y(x). (19) Put y = n Z α nx n in (19); by checking the corresponding coefficients, one gets: 1 α n = 1 (1 q)q n a n 1, n Z. It follows that, if 1 q q Z, then: α n = α 0 ((1 q)q; q) n ( def = ((1 q)q1+n ; q) ((1 q)q; q) α 0 ). (20) In the rest of this paper, we suppose that 1 q q Z. It is immediate to observe that if 1 q q Z, any formal solution will be convergent. Let s denote by ĝ the Laurent series corresponding to α 0 = 1: ĝ(x) = n Z x n ((1 q)q; q) n. If n, then ((1 q)q 1+n ; q) = O((q 1) n q n(n+1)/2 ); il follows that the polar part of ĝ(x) is divergent in C everywhere. Recall that the summation method described in (6) is valid for power series. Now one extends this method to the Laurent series ĝ in the following way (see (10)): ĝ(x) ˆB q;1 = 0 ψ 1 ( ; (1 q)q; q, ξ) = q n(n 1)/2 ξ n ((1 q)q; q) n n Z L [;q] q;1 = L [;q] q;1 ˆB q;1 ĝ(x) def = L [;q] q;1 0ψ 1 ( ; (1 q)q; q, ξ)(x), (21) where C \ ((q 1)q Z ). 8

9 Lemma 4.1 For all β q N, one has : 0ψ 1 ( ; β; q, x) = (q, x, q x ; q) (β, β x ; q). Proof It suffices to use Ramanujan s summation formula for 1 ψ 1 (cf [5], page 126, (5.2.1)), noticing also the fact that 0ψ 1 ( ; β; q, x) = α 1 ψ 1 (α; β; q, x α ). In particular, the following identity holds: ˆB q;1 ĝ(ξ) = θ(ξ) θ( ξ 1 q ) (q; q) ((1 q)q; q) ( ξ 1 q ; q). If n Z and C \ (q 1)q Z, one gets, from the formula (1): ˆB q;1 ĝ(q n ) = (q, 1 q ; q) ((1 q)q; q) θ() θ( 1 q ) 1 (1 q) n ( 1 q ; q) n from this and Lemma 4.1 one deduces that L [;q] q;1 ˆB q;1 ĝ(x) = = 1 θ() ((1 q)q, (1 q)q ; q) θ( x ) 0 ψ 1 ( ; 1 q ; q, (1 q)x ) (q; q) θ()θ( (1 q)x ) ((1 q)q, x; q) θ( x )θ( 1 q ). If x = q, one verifies without difficulties that L [;q] q;1 ˆB 1 q;1 ĝ(q) = ; (1 q; q) hence we are ready to state the following result. Theorem 4.2 Consider the formal solution ˆΓ(q; z) = (1 q; q) ĝ(q z ) of the equation (18). One has ˆΓ(q; z) = (1 q) n Z((1 q)q n ; q) q nz ; and the series ˆΓ(q; z) is [; q]-summable in the variable q z for all C\(q 1)q Z, with sum: L [;q] q;1 ˆB (q; q) θ()θ( (1 q) q z ) q;1ˆγ(q; z) = (1 q) (q z ; q) θ( 1 q )θ(q z ). When q tends to 1, the following it holds: q 1 L[;q] q;1 ˆB q;1ˆγ(q; z) = Γ(z) for all z C \ {0, 1, 2,...}, the convergence being uniform on every compact subset of C \ {0, 1, 2,...}. 9

10 Proof It only remains to check the it, which can be deduced from the following formulas (see [1]): q 1 (q; q) (q a ; q) (1 q) 1 a = Γ(a), q 1 θ(q a x) θ(q b x) = xb a, θ((1 q)q a x) q 1 θ((1 q)q b x (1 q)a b = x b a. Acknowledgement. The author would like to thank Jacques Sauloy for his constructive suggestions. References [1] R. Askey, The q-gamma and q-beta functions, Appl. Anal. 8 (1978) [2] G.D. Birkhoff, The generalized Riemann Problem for Linear Differential Equations and the Allied Problems for Linear Difference and q-difference Equations, Proc. Am. Acad., 49 (1913) [3] G.D. Birkhoff and P.E. Güenther, Note on a canonical form from the linear q-difference system, Proc. Nat. Acad. of Sci. 27 (1941) [4] L. Di Vizio, J.-P. Ramis, J. Sauloy and C. Zhang, Equations aux q- différences, Gazette des mathématiciens (SMF), to appear. [5] G. Gasper and M. Rahman, Basic hypergeometric series, Encycl. Math. Appl., Cambridge Univ, Press, Cambridge, [6] J.-P. Ramis, About the growth of entire functions solutions of linear algebraic q-difference equations, Annales de la Fac. de Toulouse, Série 6, Vol. I, no. 1(1992) [7] J.-P. Ramis J. Sauloy et C. Zhang, Classification analytique locale des équations aux q-différences, in preparation (2003). [8] J.-P. Ramis and C. Zhang, Développement asymptotique q-gevrey et fonction thêta de Jacobi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) [9] C. Zhang, Transformations de q-borel-laplace au moyen de la fonction thêta de Jacobi, C. R. Acad. Sci. Paris, Série I 331 (2000) [10] C. Zhang, Sur la fonction q-gamma de Jackson, Aequationes Math. 62 (2001) [11] C. Zhang, Une sommation discrète pour des équations aux q-différences linéaires et à coefficients analytiques: théorie générale et exemples, Proceedings of the Conference on Differential Equations and the Stokes Phenomenon (Groningen, May 2001) , ed. by B.L.J. Braaksma, G.K. Immink, M. Van der Put and J. Top, World Scientific,