Some Combinatorial and Analytical Identities

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• 作者：Mourad E. H. Ismail (1) (2)
  Dennis Stanton (3)
  
• 关键词：05A19 ; 33D15 ; 05A30 ; 33D70 ; partitions ; identities of Chen and Liu ; Dilcher ; Fu and Lascoux ; Prodinger and Uchimura ; Summation theorems ; polynomial expansions ; bibasic sums ; Watson transformation ; the Gasper identity ; Lagrange type interpolation
• 刊名：Annals of Combinatorics
• 出版年：2012
• 出版时间：December 2012
• 年：2012
• 卷：16
• 期：4
• 页码：755-771
• 参考文献：1. Alladi K.: A combinatorial study and comparison of partial theta identities of Andrews and Ramanujan. Ramanujan J. 23(1-3), 227鈥?41 (2010) CrossRef
  2. Andrews G.E.: On the foundations of combinatorial theory V, Eulerian differential operators. Stud. Appl. Math. 50, 345鈥?75 (1971)
  3. Andrews G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)
  4. Andrews, G.E.: Problems and prospects for basic hypergeometric series. In: Askey, R. (ed.) Theory and Application of Special Functions, pp. 191鈥?24. Academic Press, New York (1975)
  5. Andrews G.E., Askey R.A., Roy R.: Special Functions. Cambridge University Press, Cambridge (1999)
  6. Askey, R.,Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54(319) (1985)
  7. Chen, W.Y.C., Liu, E.: A Franklin type involution for squares. Adv. Appl. Math. (to appear)
  8. Cooper S.: The Askey-Wilson Operator and the ${_{6}\phi_{5}}$ Summation Formula. Massey University, Palmerston (1996)
  9. Davis P.J.: Interpolation and Approximation. Dover Publications, New York (1975)
  10. Dilcher K.: Some / q-series identities related to divisor function. Discrete Math. 145(1-3), 83鈥?3 (1995) CrossRef
  11. Fu A.M., Lascoux A.: / q-identities from Lagrange and Newton interpolation. Adv. Appl. Math. 31(3), 527鈥?31 (2003) CrossRef
  12. Fu A.M., Lascoux A.: / q-identities related to overpartitions and divisor functions. Electron. J. Combin. 12,鈥?R38 (2005)
  13. Gasper, G.: Elementary derivations of summation and transformation formulas for / q-series. In: Ismail, M.E.H., Masson, D.R., Rahman, M. (eds.) Special Functions, / q-Series and Related Topics, Fields Inst. Commun., 14, pp. 55鈥?0. Amer. Math. Soc., Providence (1997)
  14. Gasper G., Rahman M.: Basic Hypergeometric Series. 2nd Edit., Cambridge University Press, Cambridge (2004) CrossRef
  15. Goldman J., Rota G.C.: On the foundations of combinatorial theory IV: finite dimensional vector spaces and Eulerian generating functions. Stud. Appl. Math. 49, 239鈥?58 (1970)
  16. Guelfond A.O.: Calcul Des Diff茅rence Finies. Dunod, Paris (1963)
  17. Ismail, M.E.H.: The Askey-Wilson operator and summation theorems. In: Ismail, M., Nashed, M.Z., Zayed, A., Ghaleb, A. (eds.) Mathematical Analysis, Wavelets, and Signal Processing, Contemporary Mathematics, Vol. 190, pp. 171鈥?78. Amer. Math. Soc., Providence (1995)
  18. Ismail M.E.H.: Classical and Quantum Orthogonal Polynomials in one variable. Cambridge University Press, Cambridge (2009)
  19. Ismail M.E.H., Stanton D.: Applications of / q-Taylor theorems. J. Comput. Appl. Math. 153(1-2), 259鈥?72 (2003) CrossRef
  20. Ismail M.E.H., Stanton D.: / q-Taylor theorems, polynomial expansions, and interpolation of entire functions. J. Approx. Theory 123(1), 125鈥?46 (2003) CrossRef
  21. Ismail, M.E.H., Stanton, D.: / q-Taylor series and interpolation. preprint (2010)
  22. Ismail, M.E.H., Stanton, D.: Rodrigues type formulas, asymptotics, and biorthogonality. preprint (2010)
  23. Kirchenhofer P.: A note on alternating sums. Electron. J. Combin. 3(2), #R7 (1996)
  24. Prodinger H.: Some applications of the / q-Rice formula. Random Structures Algorithms 19(3-4), 552鈥?57 (2001) CrossRef
  25. Rahman M.: An integral representation of a ${_{10}\phi_{9}}$ and continuous bi-orthogonal ${_{10}\phi_{9}}$ rational functions. Canad. J. Math. 38(3), 605鈥?18 (1986) CrossRef
  26. Riordan J.: Combinatorial Identities. JohnWiley& Sons, Inc., New York-London-Sydney (1968)
  27. Uchimura K.: A generalization of identities for the divisor generating function. Utilitas Math. 25, 377鈥?79 (1984)
  28. Van Hamme L.: Advanced Problem 6407. Amer. Math. Month. 89(9), 703鈥?04 (1982) CrossRef
  
• 作者单位：Mourad E. H. Ismail (1) (2)
  Dennis Stanton (3)
  
  1. Department of Mathematics, University of Central Florida, Orlando, Florida, 32816, USA
  2. Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia
  3. School of Mathematics, College of Science and Engineering, University of Minnesota, Minneapolis, MN, 55455, USA
  
• ISSN：0219-3094

文摘

We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced ${_{4}\phi_{3}}$ to a very-well-poised ${_{8}\phi_{7}}$ is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials. By applying the Leibniz formula for the Askey-Wilson operator we also establish the ${_{8}\phi_{7}}$ summation theorem.