By George Gasper
This up to date version will proceed to satisfy the wishes for an authoritative entire research of the swiftly turning out to be box of uncomplicated hypergeometric sequence, or q-series. It contains deductive proofs, workouts, and helpful appendices. 3 new chapters were further to this variation overlaying q-series in and extra variables; linear- and bilinear-generating services for easy orthogonal polynomials; and summation and transformation formulation for elliptic hypergeometric sequence. additionally, the textual content and bibliography were increased to mirror fresh advancements. First version Hb (1990): 0-521-35049-2
Read or Download Basic hypergeometric series PDF
Similar geometry and topology books
Low-dimensional geometry: From Euclidean surfaces to hyperbolic knots
The research of three-dimensional areas brings jointly components from numerous components of arithmetic. the main impressive are topology and geometry, yet components of quantity thought and research additionally make appearances. some time past 30 years, there were extraordinary advancements within the arithmetic of three-dimensional manifolds.
- Elements de geometrie algebrique
- Baecklund-Transformationen und Integrabilitaetsbedingungen
- The Topology of the 2x2 Games: A New Periodic Table
- The Riemann Approach to Integration: Local Geometric Theory
- Differential Geometry Peniscola 1985
Extra resources for Basic hypergeometric series
Sample text
2) which clearly approaches n! as q ---+ 1-. Hence f q(n + 1) tends to f( n + 1) = n! as q ---+ 1-. 1). 3) q---+I we shall give a simple, formal proof due to Gosper; see Andrews [1986]. 11]) and the well-known functional equation for the gamma function r(x + 1) = xr(x), r(1) = 1. 4) For a rigorous justification of the above steps see Koornwinder [1990]. 5) f(x + 1) = - 1 - f (x), f(l) = 1. 1) is a natural q-analogue of f(x). 1) that fq(x) has poles at x = 0, -1, -2, .... The residue at x = -n is lim (x x---+-n + n) f q(x) (1 _q)n+1 x+n lim (1 - q-n)(1 - ql-n) ...
M r )] r+ 2 'l-'r+ 1 b b b ' q, a q q, 1,···, r = (q, bq/a; q)oo (bl /b; q)ml ... (br/b; q)mr bm1+ ... + mr (bq, q/a; q)oo (b l ; q)ml ... 1). 6) by replacing a, b, bl , ... , br by qa, qb, qb 1, ... , qb r , respectively, and letting q ----7 1. 6) gives '" [a,blqml, ... ,brqmr. -I -cm1+ ... +mr)] _ 0 I -I -Cm1+···+mr)1 r+ I '/'r bI, ... 7) while letting b ----700 in the case a '" r+I'/'r = q-Cml+ ... +mr) ' bI qml , ... , br qmr 1 bI, ... , br ;q, (_I)ml+···+mr(. 8) bl ; q ml ... 3). 9) when la-Iql-Cm1+···+mr)1 < 1.
2). Also see Andrews [1973]. 4) can be reversed (by replacing k by n - k) to yield Sears' [1951c] transformation formula 2¢1 = (q-n, b; c; q, z) (~jb; r)n c;q n (bZ)n 3¢2(q-n, qjz, c-1ql-n; bc-1ql-n, 0; q, q). 1). First, set c = bzq! 1) and letting a ---4 00 qn 2 zn ~ (q,zq;q)n 1 (zq; q)oo . , q! 2) (Zq2; q)oo. 8) n= 1 in terms of infinite products. 11) n= 1 and 00 194 (X, q) = II (1- q2n)(1_ 2q2n-1 cos2x + q4n-2). 12) n= 1 It is common to write 19 k (x) for 19 k (x, q), k = 1, ... ,4. 13) one can think of the theta functions 19 1 (x, q) and 192 (x, q) as one-parameter deformations (generalizations) of the trigonometric functions sin x and cos x, 1.