By Sergei Suslov

ISBN-10: 1441952446

ISBN-13: 9781441952448

ISBN-10: 1475737319

ISBN-13: 9781475737318

It was once with the book of Norbert Wiener's publication ''The Fourier In tegral and sure of Its purposes" [165] in 1933 by way of Cambridge Univer sity Press that the mathematical neighborhood got here to gain that there's another method of the research of c1assical Fourier research, particularly, throughout the thought of c1assical orthogonal polynomials. Little could he comprehend at the moment that this little thought of his could aid bring in a brand new and exiting department of c1assical research referred to as q-Fourier research. makes an attempt at discovering q-analogs of Fourier and different comparable transforms have been made through different authors, however it took the mathematical perception and instincts of none different then Richard Askey, the grand grasp of exact capabilities and Orthogonal Polynomials, to determine the traditional connection among orthogonal polynomials and a scientific conception of q-Fourier research. The paper that he wrote in 1993 with N. M. Atakishiyev and S. ok Suslov, entitled "An Analog of the Fourier rework for a q-Harmonic Oscillator" [13], used to be most likely the 1st major book during this sector. The Poisson k~rnel for the contin uous q-Hermite polynomials performs a job of the q-exponential functionality for the analog of the Fourier fundamental below considerationj see additionally [14] for an extension of the q-Fourier rework to the final case of Askey-Wilson polynomials. (Another very important factor of the q-Fourier research, that merits thorough research, is the idea of q-Fourier series.

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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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