BASIC HYPERGEOMETRIC SERIES
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1 ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS BASIC HYPERGEOMETRIC SERIES Second Edition GEORGE GASPER Northwestern University, Evanston, Illinois, USA MIZAN RAHMAN Carleton University, Ottawa, Canada CAMBRIDGE UNIVERSITY PRESS
2 Foreword V a 9 e x i Preface xxi Preface to the second edition xxv 1 Basic hypergeometric series Introduction Hypergeometric and basic hypergeometric series The (/-binomial theorem Heine's transformation formulas for 201 series Heine's (/-analogue of Gauss' summation formula Jacobi's triple product identity, theta functions, and elliptic numbers A ^-analogue of Saalschiitz's summation formula The Bailey-Daum summation formula (/-analogues of the Karlsson-Minton summation formulas The (/-gamma and (/-beta functions The ^-integral 23 Exercises 24 Notes 34 2 Summation, transformation, and expansion formulas Well-poised, nearly-poised, and very-well-poised hypergeometric and basic hypergeometric series A general expansion formula A summation formula for a terminating very-well-poised 4^3 series A summation formula for a terminating very-well-poised 6</*5 series Watson's transformation formula for a terminating very-well-poised g07 series Jackson's sum of a terminating very-well-poised balanced 807 series Some special and limiting cases of Jackson's and Watson's formulas: the Rogers-Ramanujan identities Bailey's transformation formulas for terminating 504 and 706 series Bailey's transformation formula for a terminating io09 series 47 Vll
3 viii 2.10 Limiting cases of Bailey's io0g transformation formula Bailey's three-term transformation formula for VWP-balanced 807 series Bailey's four-term transformation formula for balanced io</>9 series 55 Exercises 58 Notes 67 3 Additional summation, transformation, and expansion formulas Introduction f Two-term transformation formulas for 302 series Three-term transformation formulas for 302 series Transformation formulas for well-poised 302 and very-well-poised 504 series with arbitrary arguments Transformations of series with base q to series with base q Bibasic summation formulas Bibasic expansion formulas Quadratic, cubic, and quartic summation and transformation formulas Multibasic hypergeometric series Transformations of series with base q to series with base q 96 Exercises 100 Notes Basic contour integrals Introduction Watson's contour integral representation for 201 (fl, b; c; (/, z) series Analytic continuation of 201 (a, b; c; q, z) (/-analogues of Barnes' first and second lemmas Analytic continuation of r+i0 r series Contour integrals representing well-poised series A contour integral analogue of Bailey's summation formula Extensions to complex q inside the unit disc Other types of basic contour integrals General basic contour integral formulas 126
4 ix 4.11 Some additional extensions of the beta integral Sears' transformations of well-poised series 130 Exercises 132 Notes Bilateral basic hypergeometric series Notations and definitions Ramanujan's sum for \ij)\{a\ b; q, z) Bailey's sum of a very-well-poised gipq series A general transformation formula for an rip r series A general transformation formula for a very-well-poised 2rtp2r series Transformation formulas for very-well-poised gips ar >d IOV'IO series 145 Exercises 146 Notes The Askey Wilson (/-beta integral and some associated formulas The Askey-Wilson (/-extension of the beta integral Proof of formula (6.1.1) Integral representations for very-well-poised 807 series Integral representations for very-well-poised io09 series A quadratic transformation formula for very-well-poised balanced io09 series The Askey-Wilson integral when max (\a\, \b\, \c\, \d\) > Exercises 168 Notes Applications to orthogonal polynomials Orthogonality The finite discrete case: the (/-Racah polynomials and some special cases The infinite discrete case: the little and big g-jacobi polynomials An absolutely continuous measure: the continuous q-ultraspherical polynomials The Askey-Wilson polynomials 188
5 x 7.6 Connection coefficients A difference equation and a Rodrigues-type formula for the Askey-Wilson polynomials 197 Exercises 199 Notes Further applications Introduction A product formula for balanced 403 polynomials Product formulas for (/-Racah and Askey-Wilson polynomials A product formula in integral form for the continuous (/-ultraspherical polynomials Rogers' linearization formula for the continuous (/-ultraspherical polynomials The Poisson kernel for C n (x; /3\q) Poisson kernels for the Q-Racah polynomials (/-analogues of Clausen's formula Nonnegative basic hypergeometric series Applications in the theory of partitions of positive integers Representations of positive integers as sums of squares 242 Exercises 245 Notes Linear and bilinear generating functions for basic orthogonal polynomials Introduction The little (/-Jacobi polynomials A generating function for Askey-Wilson polynomials A bilinear sum for the Askey-Wilson polynomials I A bilinear sum for the Askey-Wilson polynomials II A bilinear sum for the Askey-Wilson polynomials III 270 Exercises 272 Notes 281
6 xi 10 g-series in two or more variables Introduction g-appell and other basic double hypergeometric series An integral representation for $W (q a ; q b, q b ' ;q c ;q; x, y) Formulas for & 2 \q a ; q b, q b '; q c, q c> ; q; x, y) Formulas for $( 3 )(<f,q a ';q b,q b '; q c ; q; x, y) Formulas for a (/-analogue of F An Askey-Wilson-type integral representation for a (/-analogue of Fi 294 Exercises 296 Notes Elliptic, modular, and theta hypergeometric series Introduction Elliptic and theta hypergeometric series Additive notations and modular series Elliptic analogue of Jackson's 807 summation formula Elliptic analogue of Bailey's transformation formula for a terminating 1009 series Multibasic summation and transformation formulas for theta hypergeometric series Rosengren's elliptic extension of Milne's fundamental theorem 331 Exercises 336 Notes 349 Appendix I Identities involving (/-shifted factorials, (/-gamma functions and (/-binomial coefficients 351 Appendix II Selected summation formulas 354 Appendix III Selected transformation formulas 359 References 367 Symbol index 415 Author index 418 Subject index 423
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