# New formulae between Jacobi polynomials and some fractional Jacobi functions generalizing some connection formulae

New formulae between Jacobi polynomials and some fractional Jacobi functions generalizing some... In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type \$\$_4F_{3}(1)\$\$ 4 F 3 ( 1 ) . With the aid of some standard reduction formulae such as Pfaff-Saalschütz’s and Watson’s identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# New formulae between Jacobi polynomials and some fractional Jacobi functions generalizing some connection formulae

, Volume 9 (1) – Jul 15, 2017
26 pages

/lp/springer-journals/new-formulae-between-jacobi-polynomials-and-some-fractional-jacobi-2YUyVa0v8p
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-017-0183-7
Publisher site
See Article on Publisher Site

### Abstract

In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type \$\$_4F_{3}(1)\$\$ 4 F 3 ( 1 ) . With the aid of some standard reduction formulae such as Pfaff-Saalschütz’s and Watson’s identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jul 15, 2017