By Sergei Suslov

It used to be with the booklet of Norbert Wiener's booklet ''The Fourier In tegral and sure of Its purposes" [165] in 1933 through Cambridge Univer sity Press that the mathematical group got here to gain that there's an alternate method of the research of c1assical Fourier research, specifically, during the conception of c1assical orthogonal polynomials. Little could he recognize at the moment that this little concept of his might support 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 similar 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 specific features and Orthogonal Polynomials, to work out the average connection among orthogonal polynomials and a scientific idea 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], was once most likely the 1st major ebook during this region. 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 crucial below considerationj see additionally [14] for an extension of the q-Fourier remodel to the overall case of Askey-Wilson polynomials. (Another very important aspect of the q-Fourier research, that merits thorough research, is the idea of q-Fourier series.

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2. First Proof of Addition Theorem: Analytic Functions We shall follow here the idea of one of the original proofs [140] using arguments of the theory of analytic funetions. 16). Also, the q-exponential funetions cq (x; ±a) are linearly independent when a f. O. 4 of this ehapter, cq (x, y; a) = A cq (x; a) + B cq (x; -al , 47 S. K. 1) 3. ADDITION THEOREMS 48 where A = A(z, s) and B = B(z, s) are, generally speaking, some functions of period 1 in z. p = qS. 2) eq (x,Yja) = A eq (xja) - B eq (Xj -a), or eq (x, Yj a) = A eq (Xj a) .

1) 0. ( -qx/y, -qy/x, 0 +-1(x + y) 31P3 -q -q, q3/2 ,-q3/2 j q, 00 n(n-1)/4 q (_q(l-n)/2 y jx jq) (ax)n n=O (q; q)n n =I: as a q-version of the expa (x + y) on a q-linear grid. This funetion has appeared in [118]. Let us introduee also cq(ax) = cq(x,O;a) = 11P1 0. 2) 2 3; q , q n(n-1)/4 = L: q 00 (0 ( (qjq)n (axt as an analog of the exp (ax) ; see [17], [38], [118]. The basie trigonometrie funetions on a q-linear grid ean be introdueed by letting 0. 5. 5) _ql/2X/y, _ql/2 y/ x , 0 1/2 1/2 -q, q ,-q X3IP3 ( q, q ; 1/2 2 w xy ) are q-analogs of the cos w (X + y) and sin w (x + y) on a q-linear grid, respectively.

Sin(wx + xy) = weos (wx + xy), ~ eos (wx + xy) = -xsin(wx + xy), ~ sin (wx + xy) = xeos (wx + xy) , respeetively. Show that Gq (Xi W) Cq (Yi x) 1 = 2" (Cq (x, Yi W, x) + Cq (x, -YiW, x)), 1 Bq (XiW) Bq (Yi x) = 2" (Cq (x, -YiW, x) - Cq (X,YiW, x)), 1 Bq (Xi w) Cq (Yi x) = 2" (Bq (x, Yi w, x) + Bq (x, -Yi w, x)) , Cq (Xi w) Bq (Yi x) = 2" (Bq (x, Yi w, x) - 1 Bq (x, -Yi w, x)) . ] CHAPTER 3 Addition Theorems In this ehapter we shall diseuss q-analogs of the addition formula for the exponential function and the eorresponding results on q-trigonometry.