By Sergei Suslov

ISBN-10: 1441952446

ISBN-13: 9781441952448

It used to be with the booklet of Norbert Wiener's e-book ''The Fourier In tegral and sure of Its purposes" [165] in 1933 by means of Cambridge Univer sity Press that the mathematical group got here to achieve that there's an alternate method of the examine of c1assical Fourier research, specifically, during the conception of c1assical orthogonal polynomials. Little might he understand at the moment that this little inspiration of his may support herald 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 by way of different authors, however it took the mathematical perception and instincts of none different then Richard Askey, the grand grasp of specified services and Orthogonal Polynomials, to determine the typical connection among orthogonal polynomials and a scientific thought of q-Fourier research. The paper that he wrote in 1993 with N. M. Atakishiyev and S. okay Suslov, entitled "An Analog of the Fourier rework for a q-Harmonic Oscillator" [13], used to be most likely the 1st major ebook during this quarter. The Poisson k~rnel for the contin uous q-Hermite polynomials performs a task of the q-exponential functionality for the analog of the Fourier imperative lower than considerationj see additionally [14] for an extension of the q-Fourier rework to the overall case of Askey-Wilson polynomials. (Another very important aspect of the q-Fourier research, that merits thorough research, is the speculation of q-Fourier series.

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**Example text**

Y (n - k)! VXI (z - n/2 + k) x TI~=o VXI (z _ (m _ k) /2) 1 (z - n + k), = 'Y (1) 'Y (2) ... 'Y (n) and 'Y (JJ) is defined in Ex. 3 [139]. [Hint: Use the mathematical induction or Cauchy's integral where 'Y (n)! formula and the identity v(n) ( z X n (8) 1 - X n (z) 'Y (n)! ] (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. 18). 15) with x (z) = (cf + q-Z) /2 = coso, qZ = ei8 ; see Ex. 11. 11). 5, Eq. 1). 1). 18) (a2/ß2;q)kqk2/4 (ßei'P)k (q; q)k . xL00 qn(n-2k)/4 ßne-m'P (q; q)n n=O X ( _q(l- n+k)/2 ei8+ i'P a/ ß, _q(l-n+k)/2ei'P-i8 a/ ß; q) n . 1) onee again. The seeond sum ean be redueed to the sum of two 4CP3 series similar to those in 20 2. 17) is an analog of exp (ax + ßy). im Eq (x, Yj (1 - q) a/2, (1 - q) ß/2) q-tl- = f: (ß/~)n n=O n. e- inrp (1+e i8 +irp a/ß)n (1+e irp - i8a/ß)n n (-n)k ( x {; ~ =L (ß/2)n n!

15) Integral representations. Let p (z) satisfy the Pearson equation [lT (z) P (z)]' = r (z) p (z) and let v be a root of the equation 1 A + vr' + 2"v (v -1) lT" = O. L = Vj (b) the contour C is chosen so that the equality lT"'(s)p(s) 181 =0 (s - z)"'+1 82 holds, where SI and s2 are the end points of the contour C. 6. EXERCISES FOR CHAPTER 2 37 (16) Power series method. Let a he a root of the equation u (z) = O. Show that the differential equation of hypergeometric type has a power series solution of the form 00 y(z) = LCn(z-a)n, n=O where Cn+l c;: = if: dk (a) lim d m-too x kYm A + n (T' + (n -1)u" /2) (n + 1) (T (a) + nq' (a» , dk (z) = d x kY (z) for k = 0,1,2; (h) lim (A - Am) Cm (z - alm = 0; m-too withYm(z) = L:'=oCn(z-at ,Am = -mr'-m(m-1)u"/2.