Askey–Wilson operator on entire functions of exponential type

Li, Xin; Ranasinghe, Rajitha

doi:10.1090/proc/14080

Askey–Wilson operator on entire functions of exponential type
HTML articles powered by AMS MathViewer

by Xin Li and Rajitha Ranasinghe PDF

Proc. Amer. Math. Soc. 146 (2018), 4283-4292 Request permission

Abstract:

In this paper, we first establish a series representation formula for the Askey–Wilson operator applied on entire functions of exponential type and then demonstrate its power in discovering summation formulas, some known and some new.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, DOI 10.1090/memo/0319
Bruce C. Berndt, Ramanujan’s notebooks. Part V, Springer-Verlag, New York, 1998. MR 1486573, DOI 10.1007/978-1-4612-1624-7
S. Bernstein, Sur une propriété des fonctions entières, C. R. Acad. Sci. Paris 176 (1923), 1603–1605.
R. P. Boas, The Derivative of a Trigonometric Integral, J. London Math. Soc. 12 (1937), no. 3, 164–165. MR 1575061, DOI 10.1112/jlms/s1-12.2.164
Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
B. Malcolm Brown and Mourad E. H. Ismail, A right inverse of the Askey-Wilson operator, Proc. Amer. Math. Soc. 123 (1995), no. 7, 2071–2079. MR 1273478, DOI 10.1090/S0002-9939-1995-1273478-X
P. L. Butzer, G. Schmeisser, and R. L. Stens, The classical and approximate sampling theorems and their equivalence for entire functions of exponential type, J. Approx. Theory 179 (2014), 94–111. MR 3148887, DOI 10.1016/j.jat.2013.11.010
R. William Gosper, Mourad E. H. Ismail, and Ruiming Zhang, On some strange summation formulas, Illinois J. Math. 37 (1993), no. 2, 240–277. MR 1208821
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 8th ed., Elsevier/Academic Press, Amsterdam, 2015. Translated from the Russian; Translation edited and with a preface by Daniel Zwillinger and Victor Moll; Revised from the seventh edition [MR2360010]. MR 3307944
J. R. Higgins, G. Schmeisser, and J. J. Voss, The sampling theorem and several equivalent results in analysis, J. Comput. Anal. Appl. 2 (2000), no. 4, 333–371. MR 1793189, DOI 10.1023/A:1010164717587
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey; Reprint of the 2005 original. MR 2542683
M. Riesz, Eine trigonometrische interpolationsformel und einige ungleichungen fúr polynome, Jahresbericht der Deutschen Mathematiker Vereinigung 23 (1914), 354–368.
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
Ahmed I. Zayed, A proof of new summation formulae by using sampling theorems, Proc. Amer. Math. Soc. 117 (1993), no. 3, 699–710. MR 1116276, DOI 10.1090/S0002-9939-1993-1116276-8