The power collection method for connection relations: Meixner polynomials

We introduce the power collection method for easily deriving connection relations for certain hypergeometric orthogonal polynomials in the $(q-)$Askey scheme. We summarize the full-extent to which the power collection method may be used. As an example, we use the power collection method to derive connection and connection-type relations for Meixner and Krawtchouk polynomials. These relations are then used to derive generalizations of generating functions for these orthogonal polynomials. The coefficients of these generalized generating functions are in general, given in term of multiple hypergeometric functions. From derived generalized generating functions, we derive corresponding contour integral and infinite series expressions by using orthogonality.

[1]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[2]  Wolfram Koepf,et al.  Representations of q-orthogonal polynomials , 2012, J. Symb. Comput..

[3]  Stéphanie Perron,et al.  The connection. , 2012, Canadian family physician Medecin de famille canadien.

[4]  R. Askey Orthogonal Polynomials and Special Functions , 1975 .

[5]  Erik Koelink,et al.  Review of ``Classical and quantum orthogonal polynomials in one variable. "by Mourad E.H. Ismail , 2011 .

[6]  M. Ismail,et al.  Expansions in the Askey{Wilson Polynomials , 2015 .

[7]  H. Volkmer,et al.  Generalizations of generating functions for higher continuous hypergeometric orthogonal polynomials in the Askey scheme , 2014, 1405.1918.

[8]  R. Askey,et al.  MULTIPLE GAUSSIAN HYPERGEOMETRIC SERIES (Ellis Horwood Series Mathematics and Its Applications) , 1986 .

[9]  Wolfram Koepf,et al.  Representations of orthogonal polynomials , 1997 .

[10]  Leon M. Hall,et al.  Special Functions , 1998 .

[11]  M. Ismail,et al.  Connection relations and expansions , 2011 .

[12]  W. Koepf,et al.  Duplication coefficients via generating functions , 2007 .

[13]  M. Ismail,et al.  Classical and Quantum Orthogonal Polynomials in One Variable: Bibliography , 2005 .

[14]  Howard S. Cohl,et al.  Generalizations of generating functions for hypergeometric orthogonal polynomials with definite integrals , 2013 .

[15]  N. Temme Special Functions: An Introduction to the Classical Functions of Mathematical Physics , 1996 .

[16]  Plamen Simeonov,et al.  Formulas and identities involving the Askey-Wilson operator , 2016, Adv. Appl. Math..

[17]  D. D. Tcheutia,et al.  Coefficients of multiplication formulas for classical orthogonal polynomials , 2016 .

[18]  Wolfram Koepf,et al.  Connection and linearization coefficients of the Askey-Wilson polynomials , 2013, J. Symb. Comput..

[19]  P. W. Karlsson,et al.  Multiple Gaussian hypergeometric series , 1985 .

[20]  Extensions of discrete classical orthogonal polynomials beyond the orthogonality , 2007, 0710.4930.

[21]  George Gasper,et al.  Projection formulas for orthogonal polynomials of a discrete variable , 1974 .

[22]  Rene F. Swarttouw,et al.  Hypergeometric Orthogonal Polynomials , 2010 .

[23]  R. S. Costas-Santos,et al.  Generalizations of generating functions for basic hypergeometric orthogonal polynomials , 2014, Open Journal of Mathematical Sciences.