# Recurrences and explicit formulae for the expansion and connection coefficients in series of the product of two classical discrete orthogonal polynomials

Document Type: Research Paper

Author

1 ‎Department of Mathematics‎, ‎Faculty of Industrial Education‎, ‎Helwan‎ ‎University‎, ‎Cairo-Egypt

2 ‎Department of Mathematics‎, ‎Faculty of Sciences‎, ‎Saqraa University‎, ‎Shaqraa-KSA.

Abstract

Suppose that for an arbitrary function $f(x,y)$ of two discrete variables, we have the formal expansions. [f(x,y)=sumlimits_{m,n=0}^{infty }a_{m,n},P_{m}(x)P_{n}(y),]
$$‎ ‎x^{m}P_{j}(x)=\sum\limits_{n=0}^{2m}a_{m,\,n}(j)P_{j+m-n}(x)‎,$$
‎we find the coefficients $b_{i,j}^{(p,q,\ell‎ ,‎\,r)}$ in the expansion‎
$$‎ ‎x^{\ell }y^{r}\,\nabla _{x}^{p}\nabla _{y}^{q}\,f(x,y)=x^{\ell‎ ‎}y^{r}f^{(p,q)}(x,y) =\sum\limits_{m,n=0}^{\infty‎ ‎}a_{m,n}^{(p,q)}\,P_{m}(x)P_{n}(y),\,\,a_{m,n}^{(0,0)}=a_{m,n}‎,$$
‎We give applications of these results in solving partial difference‎ ‎equations with varying polynomial coefficients‎, ‎by reducing them to‎ ‎recurrence relations (difference equations) in the expansion‎ ‎coefficients of the solution‎.

Keywords

Main Subjects

### References

H.M. Ahmed, Recurrence relation approach for expansion and connection coefficients in series of classical discrete orthogonal polynomials, Integral Transforms Spec. Funct. 20 (2009), no. 1, 23--34.

H.M. Ahmed and SI. El-Soubhy, Recurrences and explicit formulae for the expansion and connection coefficients in series of ordinary Bessel polynomials, Appl. Math. Comput. 199 (2008) 482--493.

R. Alvarez-Nodarse, R.J. Ya~nez and J.S. Dehesa, Modified Clebsch-Gordon-type expansions for products of discrete hypergeometric polynomials, J. Comput. Appl. Math. 89 (1998) 171--197.

I. Area, N. Atakishiyev, E. Godoy and J. Rodal, Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions, Appl. Math. Comput. 223 (2013) 520--536.

I. Area and E. Godoy, On limit relations between some families of bivariate hypergeometric orthogonal polynomials, J. Phys. A 46 (2013), 035202, 11 pages.

I. Area, E. Godoy and J. Rodal, On a class of bivariate second-order linear partial difference equations and their monic orthogonal polynomial solutions, J. Math. Anal. Appl. 389 (2012), no. 1, 165--178.

I. Area, E. Godoy, J. Rodal, A. Ronveaux and A. Zarzo, Bivariate Kravchuk polynomials: Inversion and connection problems with the NAVIMA algorithm, J. Comput. Appl. Math. 284 (2015) 50--57.

I. Area, E. Godoy, A. Ronveaux and A. Zarzo, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Discrete case, J. Comput. Appl. Math. 89 (1998) 309--325.

I. Area, E. Godoy, A. Ronveaux and A. Zarzo, A. Bivariate second-order linear partial differential equations and orthogonal polynomial solutions, J. Math. Anal. Appl. 387 (2012), no. 2, 1188--1208.

E.H. Doha, The Chebyshev coefficients of general-order derivatives of an infinitely differentiable function in two or three variables, Ann. Univ. Sci. Budapest. Sect. Comput. 13 (1992) 83--91.

E.H. Doha, On the coefficients of differentiated expansions of double and triple Legendre polynomials, Ann. Univ. Sci. Budepest. Sect. Comput. 15 (1995) 25--35.

E.H. Doha, The coefficients of differentiated expansions of double and triple ultraspherical polynomials, Ann. Univ. Sci. Budepest. Sect. Comput. 19 (2000) 57--73.

E.H. Doha, On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials, J. Phys. A 36 (2003) 5449--5462.

E.H. Doha, On the connection coefficients and recurrence relations arising from expansions in series of Hermite polynomials, Integral Transforms Spec. Funct. 15 (2004), no. 1, 13--29.

E.H. Doha, On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. Phys. A 37 (2004) 657--675.

E.H. Doha, W.M. Abd-Elhameed and H.M. Ahmed, The coefficients of differentiated expansions of double and triple Jacobi polynomials, Bull. Iranian Math. Soc. 38 (2012), no. 3, 739--766.

E.H. Doha and H.M. Ahmed, Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomials, J. Phys. A 37 (2004) 8045--8063.

E.H. Doha and H.M. Ahmed, Recurrences and explicit formulae for the expansion and connection coefficients in series of classical discrete orthogonal polynomials, Integral Transforms Spec. Funct. 17 (2006), no. 5, 329--353.

E.H. Doha and H.M. Ahmed, Recurrence relation approach for expansion and connection coefficients in series of Hahn polynomials, Integral Transforms Spec. Funct. 17 (2006), no. 11, 785--801.

I.M. Gel'fand, R.A. Minlos and Z.Ya. Sapiro, Representations of the Rotation Group and of the Lorentz Group and Their Applications, MacMillan, NewYork, 1963.

E. Godoy, A. Ronveaux, A. Zarzo and I. Area, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case, J. Comput. Appl. Math. 84 (1997) 257--275.

E. Godoy, A. Ronveaux, A. Zarzo and I. Area, On the limit relation between classical continuous and discrete orthogonal polynomials, J. Comput. Appl. Math. 91 (1998) 97--105.

E. Godoy, A. Ronveaux, A. Zarzo and I. Area, Connection problems for polynomial solution of nonhomogeneous differential and difference equations, J. Comput. Appl. Math. 99 (1998) 177--187.

R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer, Berlin, 2010.

W. Koepf and D. Schmersau, Representations of orthogonal polynomials, J. Comput. Appl. Math. 90 (1998) 57--94.

S. Lewanowicz, Second-order recurrence relations for the linearization coefficients of the classical orthogonal polynomials, J. Comput. Appl. Math. 69 (1996) 159--170.

S. Lewanowicz, Recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable, J. Comput. Appl. Math. 76 (1996) 213--229.

S. Lewanowicz and P. Wozny, Algorithms for construction of recurrence relations for the coefficients of expansions in series of classical orthogonal polynomials, Preprint, Inst. of Computer Sci. Univ. of Wroclaw, Feb. 2001, webpage http://www.ii.uni.wroc.pl/sle/publ.html.

M. Lorente, Orthogonal polynomials of several discrete variables and the 3nj-Wigner symbols: applications to spin networks, Communication presented to the XXIV International Congress on Group Theoretical Methods in Physics Paris, July 2002, Arxiv:math-ph/0402007.

Y.L. Luke, The Special Functions and Their Approximations, Academic Press, New York, 1969.

Mathematica Version 8.0, Wolfram Research 2010 Inc. (Wolfram Research Champaign).

A.F. Nikiforov, S.K. Suslov and V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Verlag, Berlin, 1991.

J. Rodal, I. Area and E. Godoy, Orthogonal polynomials of two discrete variables on the simplex, Integral Transforms Spec. Funct. 16 (2005), no. 3, 263--280.

J. Rodal, I. Area and E. Godoy, Linear partial difference equations of hypergeometric type: Orthogonal polynomial solutions in two discrete variables, J. Comput. Appl. Math. 200 (2007) 722--748.

J. Rodal, I. Area and E. Godoy, Structure relations for monic orthogonal polynomials in two discrete variables, J. Math. Anal. Appl. 340 (2008), no. 2, 825--844.

A. Ronveaux, I. Area, E. Godoy and A. Zarzo, Lectures on recursive approach to connection and linearization coefficients between polynomials, in: Proceedings Int. Workshop on Special Functions and Differential Equations, pp. 13--24, Madras, India, 1997.

A. Ronveaux, S. Belmehdi, E. Godoy and A. Zarzo, Recurrence relation approach for connection coefficients. Applications to classical discrete orthogonal polynomials, in:

Symmetries and Integrability of Difference Equations (Estrel, PQ, 1994), pp. 319--335, CRM Proc. Lecture Notes, 9, Amer. Math. Soc., Providence, RI, 1996.

A. Ronveaux, A. Zarzo and E. Godoy, Recurrence relations for connection coefficients between two families of orthogonal polynomials, J. Comput. Appl. Math. 62 (1995) 67--73.

A.V. Rozenblyum, Representations of Lie groups and multidimensional special functions, Acta Appl. Math. 29 (1992), no.3, 171--240.

J. Sanchez-Ruiz, Linearization and connection formulae involving squares of Gegenbauer polynomials, Appl. Math. Lett. 14 (2001), no. 3, 261--267.

S.K. Suslov, The 9j-symbols as orthogonal polynomials in two discrete variables, Sov. J. Nucl. Phys. 38 (1984), no. 4, 662--663.

M.V. Tratnik, Multivariable Meixner, Krawtchouk, and Meixner-Pollaczek polynomials, J. Math. Phys. 30 (1989), no. 12, 2740--2749.

M.V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau-discrete families, J. Math. Phys. 32 (1991), no. 9, 2337--2342.

J.F. Van Diejen, Properties of some families of hypergeometric orthogonal polynomials in several variables, Trans. Amer. Math. Soc. 351 (1999), no. 1, 233--270.

Y. Xu, On discrete orthogonal polynomials of several variables, Adv. Appl. Math. 33 (2004), no. 3, 615--632.

Y. Xu, Second order difference equations and discrete orthogonal polynomials of two variables, Int. Math. Res. Not. IMRN 8 (2005) 449--475.

C.-an Yu, A Constraction of general solution for a class of non-homogeneous recurrence of variable cefficients with two indices, Wuhan Univ. J. Nat. Sci. 5 (2000), no. 4, 349--85.

C.-an Yu, A formula of solution for a class of linear recurence with two indices, Wuhan Univ. J. Nat. Sci. 11 (2006), no. 3, 465--68.

C.-an Yu, A general solution for trinomial linear recurrence with two indices, Wuhan Univ. J. Nat. Sci. 16 (2011), no. 3, 190--192.

D. Yu and C.-an Yu, A formula of general solution for a class of homogeneous trinomial recurrence of variable coefficients with two indices, Wuhan Univ. J. Nat. Sci. 10 (2005), no. 5, 828-832.

P. Wozny, Recurrence relations for the coefficients of expansions in classical orthogonal polynomials of a discrete variable, Appl. Math. (Warsaw) 30 (2003) 89--107.

A. Zarzo, I. Area, E. Godoy and A. Ronveaux, Results for some inversion problems for classical continuous and discrete orthogonal polynomials, J. Phys. A 30 (1997), no. 3, L35--L40.

A. Zhedanov, 9j-symbols of the oscillator algebra and Krawtchouk polynomials in two variables, J. Phys. A 30 (1997), no. 23, 8337--8353.

### History

• Receive Date: 11 April 2017
• Revise Date: 04 January 2018
• Accept Date: 05 January 2018