On finite sums of Laguerre polynomials. (English) Zbl 1214.33004

The author considers finite sums of Laguerre polynomials and writes them in terms of Laguerre polynomials themselves or as confluent or other hypergeometric functions. He gives alternative approaches using contour integration and other integral representations. He also finds a Kummer-like transformation for a particular generalized hypergeometric function \(_{2}F_{2}\) in terms of Laguerre polynomials.


33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
33C20 Generalized hypergeometric series, \({}_pF_q\)
Full Text: DOI


[1] G.E. Andrews, R. Askey and R. Roy, Special functions , Cambridge University Press, Cambridge, 1999. · Zbl 0920.33001
[2] W.N. Bailey, Generalized hypergeometric series , Cambridge University Press, Cambridge, 1935. · Zbl 0011.02303
[3] C. Berenstein, D.-C. Chang and J. Tie, Laguerre calculus and its applications on the Heisenberg group , American Mathematical Society, Philadelphia, 2001. · Zbl 0986.22008
[4] T.S. Chihara, An introduction to orthogonal polynomials , Gordon & Breach, New York, 1978. · Zbl 0389.33008
[5] M.W. Coffey, The theta-Laguerre calculus formulation of the Li/Keiper constants , J. Approx. Theory 146 (2007), 267-275. · Zbl 1120.33006 · doi:10.1016/j.jat.2006.10.006
[6] —, Polygamma theory, the Li/Keiper constants, and the Li criterion for the Riemann hypothesis , Rocky Mountain J. Math., · Zbl 1226.11085 · doi:10.1216/RMJ-2010-40-6-1841
[7] —, Properties and possibilities of quantum shapelets , J. Phys. A 39 (2006), 897-887. · Zbl 1088.94003 · doi:10.1088/0305-4470/39/4/009
[8] A. Erdélyi et al., eds., Higher transcendental functions , Vols. 1 and 2, McGraw-Hill, Malabar, Florida, 1953. · Zbl 0052.29502
[9] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products , Academic Press, New York, 1980. · Zbl 0521.33001
[10] E.R. Hansen, A table of series and products , Prentice-Hall Inc., Englewood Cliffs, NJ, 1975. · Zbl 0438.00001
[11] X.-J. Li, The positivity of a sequence of numbers and the Riemann hypothesis , J. Number Theory 65 (1997), 325-333. · Zbl 0884.11036 · doi:10.1006/jnth.1997.2137
[12] Y.L. Luke, The special functions and their approximations , Vol. 1, Academic Press, New York, 1969. · Zbl 0193.01701
[13] G. Szegő, Orthogonal polynomials , AMS Colloq. Publ. 23 , American Mathematical Society, fourth ed., Providence, RI, 1975.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.