By Balakrishnan A.V.

ISBN-10: 0387905278

ISBN-13: 9780387905273

ISBN-10: 3540905278

ISBN-13: 9783540905271

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2. 27). 8) are well-defined when I 2 (1- q)2 q-I/2Aju l < 1. But the analytic continuation of the 2'PIS on the right side in to the larger domain can be easily given [46]. 8) one gets _l-'q1+ 2Z , _ql-2z I I-' 2'PI ( ; q ,( q (_l-'q1+2Z(, _ql-2Z(j1-'; q2) 00 PROOF.

24) Analogs 01 integral representations. Let x (z) be a nonuniform lattice of the classical type considered in Ex. 3. +I) ' and 0 is a contour in the complex z-plane, if: 42 2. BASIC EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS (a) p (z) and Pv (z) satisfy the Pearson type equations (z) P (z)] = 1" (z) P (z) VX1 (z) , [0" (z) Pv (z)] = 1"v (z) Pv (z) VXv +1 (z), ß ß [0" and 11 is a root of the equation Av + Cl( (11) l' (11) T' - l' (11 - 1) l' (11) 0:" /2 (b) the generalized powers [x (8) - (8) - [xv (z)] Xv = [Xv (8) = [Xv (8) [X II -1 (z)](~) Xv (z)](~+l) , X II -1 = [XII_~ (8 + JL) (8) - (8) - Xv (8 + 1) - = [XII [xv Xv Xv [X v -1 X = 0; (z)](~) have the properties (z -1)](~) (8) - Xv (z - JL)] (z)](~) [XII_~ (8) - XII_~ (z)] Xv-~ (z)] [X II -1 (8) - Xv (z)](~) (z)](M1) , (c) the difference-differentiation formula V'Pv~ (z) = l' (JL + 1) 'Pli, ~+1 (z) holds for JL = 11 and JL = 11 - 1; Vxv_~ (z) (d) the equations 0" [X II -1 (8) - (8) Pv (8) XII-l (z + 1)](11+1) b =0 a and { ß ( 0" (8) Pv (8) ) d8 = 0 Je [X II -1 (8) - XII-l (z + 1)](11+1) are satisfied in the cases of the sum and the integral, respectively.

3. +I) ' and 0 is a contour in the complex z-plane, if: 42 2. BASIC EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS (a) p (z) and Pv (z) satisfy the Pearson type equations (z) P (z)] = 1" (z) P (z) VX1 (z) , [0" (z) Pv (z)] = 1"v (z) Pv (z) VXv +1 (z), ß ß [0" and 11 is a root of the equation Av + Cl( (11) l' (11) T' - l' (11 - 1) l' (11) 0:" /2 (b) the generalized powers [x (8) - (8) - [xv (z)] Xv = [Xv (8) = [Xv (8) [X II -1 (z)](~) Xv (z)](~+l) , X II -1 = [XII_~ (8 + JL) (8) - (8) - Xv (8 + 1) - = [XII [xv Xv Xv [X v -1 X = 0; (z)](~) have the properties (z -1)](~) (8) - Xv (z - JL)] (z)](~) [XII_~ (8) - XII_~ (z)] Xv-~ (z)] [X II -1 (8) - Xv (z)](~) (z)](M1) , (c) the difference-differentiation formula V'Pv~ (z) = l' (JL + 1) 'Pli, ~+1 (z) holds for JL = 11 and JL = 11 - 1; Vxv_~ (z) (d) the equations 0" [X II -1 (8) - (8) Pv (8) XII-l (z + 1)](11+1) b =0 a and { ß ( 0" (8) Pv (8) ) d8 = 0 Je [X II -1 (8) - XII-l (z + 1)](11+1) are satisfied in the cases of the sum and the integral, respectively.

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