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.
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.
Year 2017,
Volume: 46 Issue: 2, 165 - 176, 01.04.2017
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.
Abd- Elhameed, W. (2017). New formulae of squares of some Jacobi polynomials via hypergeometric functions. Hacettepe Journal of Mathematics and Statistics, 46(2), 165-176.
AMA
Abd- Elhameed W. New formulae of squares of some Jacobi polynomials via hypergeometric functions. Hacettepe Journal of Mathematics and Statistics. April 2017;46(2):165-176.
Chicago
Abd- Elhameed, W.m. “New Formulae of Squares of Some Jacobi Polynomials via Hypergeometric Functions”. Hacettepe Journal of Mathematics and Statistics 46, no. 2 (April 2017): 165-76.
EndNote
Abd- Elhameed W (April 1, 2017) New formulae of squares of some Jacobi polynomials via hypergeometric functions. Hacettepe Journal of Mathematics and Statistics 46 2 165–176.
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W. Abd- Elhameed, “New formulae of squares of some Jacobi polynomials via hypergeometric functions”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 2, pp. 165–176, 2017.
ISNAD
Abd- Elhameed, W.m. “New Formulae of Squares of Some Jacobi Polynomials via Hypergeometric Functions”. Hacettepe Journal of Mathematics and Statistics 46/2 (April 2017), 165-176.
JAMA
Abd- Elhameed W. New formulae of squares of some Jacobi polynomials via hypergeometric functions. Hacettepe Journal of Mathematics and Statistics. 2017;46:165–176.
MLA
Abd- Elhameed, W.m. “New Formulae of Squares of Some Jacobi Polynomials via Hypergeometric Functions”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 2, 2017, pp. 165-76.
Vancouver
Abd- Elhameed W. New formulae of squares of some Jacobi polynomials via hypergeometric functions. Hacettepe Journal of Mathematics and Statistics. 2017;46(2):165-76.