Linearization formulae for certain Jacobi polynomials

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• 作者：W. M. Abd-Elhameed ; E. H. Doha ; H. M. Ahmed
• 关键词：Jacobi polynomials ; Linearization problems ; Generalized hypergeometric functions ; Algorithms by Zeilberger ; Petkovsek and van Hoeij
• 刊名：The Ramanujan Journal
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：39
• 期：1
• 页码：155-168
• 全文大小：432 KB
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  2.Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)MATH CrossRef
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  8.Doha, E.H.: On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J. Phys. A Math. Gen. 37, 657–675 (2004)MATH MathSciNet CrossRef
  9.Doha, E.H., Abd-Elhameed, W.M.: New linearization formulae for the products of Chebyshev polynomials of third and fourth kinds. Rocky Mt. J. Math. (To appear)
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  11.Doha, E.H., Abd-Elhameed, W.M., Ahmed, H.M.: The coefficients of differentiated expansions of double and triple Jacobi polynomials. Bull. Iran. Math. Soc. 38, 766–799 (2012)MathSciNet
  12.Fields, J.L., Wimp, J.: Expansions of hypergeometric functions in hypergeometric functions. Math. Comput. 15, 390–395 (1961)MATH MathSciNet CrossRef
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  18.Koepf, W.: Hypergeometric Summation, 2nd edn. Springer Universitext Series. http://​www.​hypergeometric-summation.​org (2014)
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  23.Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010). DLMF, Digital Library of Mathematical Functions. http://​dlmf.​nist.​gov
  24.Rahman, M.: A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Can. J. Math. 33, 915–928 (1981)MATH CrossRef
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  28.Sánchez-Ruiz, J., Artés, P.L., Martínez-Finkelshtein, A., Dehesa, J.S.: General linearization formulae for products of continuous hypergeometric-type polynomials. J. Phys. A Math. Gen. 32, 7345–7366 (1999)MATH CrossRef
  29.Tcheutia, D.D.: On connection, linearization and duplication coefficients of classical orthogonal polynomials. Ph.D. thesis, University of Kassel. https://​kobra.​bibliothek.​uni-kassel.​de/​handle/​urn:​nbn:​de:​hebis:​34-2014071645714 (2014)
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• 作者单位：W. M. Abd-Elhameed (1) (2)
  E. H. Doha (2)
  H. M. Ahmed (3) (4)
  
  1. Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia
  2. Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt
  3. Department of Mathematics, Faculty of Industrial Education, Helwan University, Cairo, Egypt
  4. Department of Mathematics, Faculty of Science, Shaqra University, Shaqra, Saudi Arabia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Number Theory
  Field Theory and Polynomials
  Combinatorics
  Fourier Analysis
  Functions of a Complex Variable
  
• 出版者：Springer U.S.
• ISSN：1572-9303

文摘

In this article, some new linearization formulae of products of Jacobi polynomials for certain parameters are derived. These new derived formulae are expressed in terms of hypergeometric functions of unit argument, and they generalize some existing formulae in the literature. With the aid of some standard formulae and also by employing symbolic algebraic computation, and in particular Zeilberger’s algorithm, several reduction formulae for summing certain terminating hypergeometric functions of unit argument are given, and hence several linearization formulae of products of Jacobi polynomials for special parameters free of hypergeometric functions are deduced.