New formulae of squares of some Jacobi polynomials via hypergeometric functions

Year 2017, Volume: 46 Issue: 2, 165 - 176, 01.04.2017

W.m. Abd- Elhameed

Abstract

In this article, a new formula expressing explicitly the squares of Jacobi polynomials of certain parameters in terms of Jacobi polynomials of arbitrary parameters is derived. The derived formula is given in terms of ceratin terminating hypergeometric function of the type $_4F_3(1)$. In
some cases, this $_4F_3(1)$ can be reduced by using some well-known reduction formulae in literature such as Watson's and Pfa-Saalschütz's
identities. In some other cases, this $_4F_3(1)$ can be reduced by means of symbolic computation, and in particular Zeilberger's, Petkovsek's and van Hoeij's algorithms. Hence, some new squares formulae for Jacobi polynomials of special parameters can be deduced in reduced forms
which are free of any hypergeometric functions.

Keywords

Jacobi polynomials, linearization coefficients, generalized hypergeometric functions, computer algebra, standard reduction formulae

References

• W.M. Abd-Elhameed. On solving linear and nonlinear sixth-order two point boundary value problems via an elegant harmonic numbers operational matrix of derivatives. CMES-Comp. Model. Eng., 101(3):159185, 2014.
• W.M. Abd-Elhameed. New formulae for the linearization coefficients of some nonsymmetric Jacobi polynomials. Adv. Dier. Eq., 2015(1):113, 2015.
• W.M. Abd-Elhameed. New product and linearization formulae of Jacobi polynomials of certain parameters. Integr. Transf. Spec, 26(8):586599, 2015.
• W.M. Abd-Elhameed, E.H. Doha, and H.M. Ahmed. Linearization formulae for certain jacobi polynomials. Ramanujan J., 39(1):155168, 2016.
• W.M. Abd-Elhameed, E.H. Doha, and Y.H. Youssri. Ecient spectral-Petrov-Galerkin methods for third-and fifth-order differential equations using general parameters generalized jacobi polynomials. Quaest. Math., 36(1):1538, 2013.
• G.E. Andrews, R. Askey, and R. Roy. Special functions. Cambridge University Press, Cambridge, 1999.
• R. Askey and G. Gasper. Linearization of the product of Jacobi polynomials. III. Can. J. Math, 23:332338, 1971.
• H. Bateman, A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi. Higher transcendental functions, volume I. McGraw-Hill New York, 1953.
• H. Chaggara and W. Koepf. On linearization coefficients of Jacobi polynomials. Appl. Math. Lett., 23(5):609614, 2010.
• E.H. Doha. On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials. J. Phys. A: Math. Gen., 36(20):54495462, 2003.
• E.H. Doha. On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J. Phys. A: Math. Gen., 37(3):657, 2004.
• E.H. Doha and W.M. Abd-Elhameed. Integrals of Chebyshev polynomials of third and fourth kinds: An application to solution of boundary value problems with polynomial coefficients. J. Contemp. Math. Anal., 49(6):296308, 2014.
• E.H. Doha and W.M. Abd-Elhameed. New linearization formulae for the products of chebyshev polynomials of third and fourth kinds. Rocky Mt. J. Math., 46(2):443460, 2016.
• E.H. Doha, W.M. Abd-Elhameed, and H.M. Ahmed. The coefficients of differentiated expansions of double and triple Jacobi polynomials. B. Iran. Math. Soc., 38(3):739766, 2012.
• 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: Math. Gen., 37(33):8045, 2004.
• K.T. Elgindy and K.A. Smith-Miles. Solving boundary value problems, integral, and integrodifferential equations using Gegenbauer integration matrices. J. Comput. Appl. Math., 237(1):307325, 2013.
• J.L. Fields and J. Wimp. Expansions of hypergeometric functions in hypergeometric functions. Math. Comp., 15(76):390395, 1961.
• G. Gasper. Linearization of the product of Jacobi polynomials I. Can. J. Math., 22:171175, 1970.
• G. Gasper. Linearization of the product of Jacobi polynomials II. Can. J. Math., 22:582 593, 1970.
• E.A. Hylleraas. Linearization of products of Jacobi polynomials. Math. Scand., 10:189200, 1962.
• W. Koepf. Hypergeometric summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.
• Y.L. Luke. The special functions and their approximations. Academic press, New York, 1969.
• C. Markett. Linearization of the product of symmetric orthogonal polynomials. Constr. Approx., 10(3):317338, 1994.
• J.C. Mason and D.C. Handscomb. Chebyshev polynomials. Chapman and Hall, New York, NY, CRC, Boca Raton, 2010.
• F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark. NIST Handbook of Mathematical functions. Cambridge University Press, 2010.
• M. Rahman. A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Can. J. Math., 33(4):915928, 1981.
• E.D. Rainville. Special Functions. The Maximalan Company, New York, 1960.
• J. Sánchez-Ruiz. Linearization and connection formulae involving squares of Gegenbauer polynomials. Appl. Math. Lett., 14(3):261267, 2001.
• J. Sánchez-Ruiz and J.S. Dehesa. Some connection and linearization problems for polynomials in and beyond the askey scheme. J. Comput. Appl. Math., 133(1):579591, 2001.
• D.D. Tcheutia. On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials. PhD thesis, Universität Kassel 2014. Available at https://kobra.bibliothek.uni-kassel.de/handle/urn:nbn:de:hebis:34-2014071645714., 2014.
• M. van Hoeij. Finite singularities and hypergeometric solutions of linear recurrence equations. J. Pure Appl. Algebra, 139(1):109131, 1999.
• G.N. Watson. A note on generalized hypergeometric series. Proc. London Math. Soc, 2:23, 1925.