Difference Hypergeometric Functions

  • N. M. Atakishiyev
  • S. K. Suslov
Conference paper
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 19)


The particular solutions for hypergeometric-type difference equations on non-uniform lattices are constructed by using the method of undetermined coefficients. Recently there has been revived interest in the classical theory of special functions. In particular, this theory has been further developed through studying their difference analogs [AWl], [AW2], [NU1], [NSU], [AS1], [AS2], and [S]. Quantum algebras [D], [VS], [KR], [K], which are being developed nowadays, provide a natural basis for the group-theoretic interpretation of these difference special functions. In the present paper we discuss a method of constructing the solutions for difference equations of hypergeometric type on non-uniform lattices [AS2].


Quantum Group Homogeneous Equation Undetermined Coefficient Basic Hypergeometric Series Hypergeometric Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [AS1]
    Atakishiyev, N.M. and Suslov, S.K. About one class of special functions, Revista Mexicana de Fisica, vol. 34, No. 2, (1988), p. 152–167.MathSciNetGoogle Scholar
  2. [AS2]
    Atakishiyev, N.M. and Suslov, S.K. Difference hypergeometric functions, and Construction of solutions of the hypergeometric-type difference equation on non-uniform lattices, Physics Institute preprints No. 319 and 323, Baku, Azerbaijan SSR, 1989 (in Russian).Google Scholar
  3. [AS3]
    Atakishiyev, N.M. and Suslov, S.K. On the moments of classical and related polynomials, Revista Mexicana de Fisica, vol. 34, No. 2, (1988), p. 147–151.MathSciNetGoogle Scholar
  4. [AW1]
    Askey, R. and Wilson, J.A., A set of orthogonal polynomials that generalize the Racah coefficients or 6j-symbols, SIAM J. Math. Anal., vol. 10, No. 5, (1979), p. 1008–1016.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [AW2]
    Askey, R. and Wilson, J.A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, 1985, Memoirs Amer. Math. Soc, No. 319.Google Scholar
  6. [Ba]
    Bailey, W., Generalized Hypergeometric Series, Cambridge: At the University Press, 1935.zbMATHGoogle Scholar
  7. [B1]
    Boole, G., A Treatise on the Calculus of Finite Differences, 2nd ed. London: Macmillan, 1872; New York: Dover, 1960, p. 236–263.Google Scholar
  8. [B2]
    Boole, G., A Treatise on Differential Equations, 5th ed., New York: Chelsea, 1959.Google Scholar
  9. [D]
    Drinfel’d, V.G., Quantum Groups, Proceedings of the International Congress of Mathematicians, Berkeley, CA, 1986: American Mathematical Society, 1987, p. 798–820.Google Scholar
  10. [EMOT]
    Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F., Higher transcendental functions, McGraw-Hill, New York, Vol. I, II, (1953), Vol. III, (1955).Google Scholar
  11. [GR]
    Gasper, G. and Rahman, M., Basic hypergeometric series, Cambridge University Press, Cambridge, 1990.zbMATHGoogle Scholar
  12. [H1]
    Hahn, W., Beitrage zur Theorie der Heineschen Reihen. Die 24 Integrale der hypergeometrischen q-Differenzengleichung. Das q-Analogon der Laplace-Transformation, Math. Nachr. Vol. 2, No. 6, (1949), p. 340–379.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [H2]
    Hahn, W., Uber Orthogonalpolynome, die q-Differenzen-gleichungen genugen, Math. Nachr., Vol. 2, No. 1, (1949), p. 4–34.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [H3]
    Hahn, W., Uber Polynome, die gleichzeitig zwei verschiedenen Orthogonalsystemen angehoren, Math. Nachr., Vol. 2, No. 5, (1949), p. 263–278.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [He1]
    Heine, E., Handbuch der Kugelfunctionen, Vol.I, Berlin: Druck und Verlag von G. Reimer, (1878), p. 97–125, 273–285.Google Scholar
  16. [He2]
    Heine, E.,Uber die Reihe \(1 + \frac{{\left( {{q^\alpha } - 1} \right)\left( {{q^\beta } - 1} \right)}}{{\left( {q - 1} \right)\left( {{q^\gamma } - 1} \right)}}.x + \frac{{\left( {{q^{\alpha + 1}} - 1} \right)\left( {{q^\alpha } - 1} \right)\left( {{q^{\beta + 1}} - 1} \right)\left( {{q^\beta } - 1} \right)}}{{\left( {{q^2} - 1} \right)\left( {q - 1} \right)\left( {{q^{\gamma + 1}} - 1} \right)\left( {{q^\gamma } - 1} \right)}}.{x^2} + \cdots \), J reine u. angew. Math, Vol. 32, No. 3,(1846), p. 210–212.zbMATHCrossRefGoogle Scholar
  17. [He3]
    Heine, E,Untersuchungen uber die Reihe \(1 + \frac{{\left( {1 - {q^\alpha }} \right)\left( {1 - {q^\beta }} \right)}}{{\left( {1 - q} \right)\left( {1 - {q^\gamma }} \right)}}.x + \frac{{\left( {1 - {q^\alpha }} \right)\left( {1 - {q^{\alpha + 1}}} \right)\left( {1 - {q^\beta }} \right)\left( {1 - {q^{\beta + 1}}} \right)}}{{\left( {1 - q} \right)\left( {1 - {q^2}} \right)\left( {1 - {q^\gamma }} \right)\left( {1 - {q^{\gamma + 1}}} \right)}}.{x^2} + \cdots \), J. reine u. angew. Math., Vol. 34, No. 4, (1847), p. 285–328.zbMATHCrossRefGoogle Scholar
  18. [ILVW]
    Ismail, M.E.H., Letessier, J., Valent, G., and Wimp, J., Two families of associated Wilson polynomials, Canad. J. Math., Vol. 42, (1990), p. 659–695.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [IR]
    Ismail, M.E.H., and Rahman, M., The associated Askey-Wilson polynomials, Trans. Amer. Math. Soc., (1991), to appear.Google Scholar
  20. [KR]
    Kirillov, A.N., and Reshetikhin, N. Yu., Representations of the algebra U q(sl(2)), q-orthogonal polynomials and invariants of links, LOMI Preprint E-9-88, Leningrad, 1988.Google Scholar
  21. [K]
    Koornwinder, T.H., Orthogonal polynomials in connection with quantum groups, Orthogonal Polynomials: Theory and Practice, ed. by P. Nevai, NATO ASI Series C, Vol. 294, Kluwer Academic Publishers, 1990, p. 257–292.Google Scholar
  22. [NR]
    Nassrallah, B. and Rahman, M., Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials, SIAM J. Math. Anal., Vol. 16, No. 1, (1985), p. 186–197.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [NSU]
    Nikiforov, A.F., Suslov, S.K., and Uvarov, V.B., Classical orthogonal polynomials of a discrete variable, Nauka, Moscow, 1985 (in Russian).zbMATHGoogle Scholar
  24. [NU1]
    Nikiforov, A.F. and Uvarov, V.B., Classical orthogonal polynomials of a dicrete variable on non-uniform lattices, Preprint No. 17, Keldysh Inst. Appl. Math., Moscow, 1983 (in Russian).Google Scholar
  25. [NU2]
    Nikiforov, A.F. and Uvarov, V.B., Special Functions of Mathematical Physics, 2nd ed., Nauka, Moscow, 1984 (in Russian).zbMATHGoogle Scholar
  26. [R]
    Rahman, M., An integral representations of a 10Φ9 and continuous bi-orthogonal 10Φ9 rational functions, Can. J. Math., Vol. 38, No. 3, (1986), p. 605–618.zbMATHCrossRefGoogle Scholar
  27. [S]
    Suslov, S.K., The theory of difference analogues of special functions of hypergeometric type, Russian Math. Surveys, The London Mathematical Society, Vol. 44, No. 2, (1989), p. 227–278, correction in ibid Vol. 45, No. 3, (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  28. [T1]
    Thomae, J.,Beitrage zur Theorie der durch Heinesche Reihe:\(1 + \frac{{1 - {q^a}}}{{1 - q}}.\frac{{1 - {q^b}}}{{1 - {q^c}}}.x + \frac{{1 - {q^a}}}{{1 - q}}.\frac{{1 - {q^{a + 1}}}}{{1 - {q^2}}}.\frac{{1 - {q^b}}}{{1 - {q^c}}}.\frac{{1 - {q^{b + 1}}}}{{1 - {q^{c + 1}}}}.{x^2} + \cdots \) darstellbaren Functionen, J. reine u. angew. Math., Vol. 70, No. 3, (1869), p. 258–281.CrossRefGoogle Scholar
  29. [T2]
    Thomae, J.,Integration der Differenzengleichung (n+æ+1)(n+λ+1)Δ2ϕ(n)+(a+bn)Δϕ(n)+cϕ(n) = 0, Zeitschrift f. Mathematik u. Physik, 1971, Vol. 16, No. 2, p. 146–158; No. 5, p. 428–439.Google Scholar
  30. [VK]
    Vaksman, L.L. and Korogodsky, L.I., Algebra of bounded functions on the quantum group of plane motions and q-analogues of Bessel functions, Dokl. Akad. Nauk SSSR, Vol. 304, (1989), p. 1036–1040 (in Russian), English translation in Soviet Mathematics.Google Scholar
  31. [VS]
    Vaksman, L.L. and Soibel’man, Algebra of functions on the quantum group SU(2), Functional Anal. Appl., Vol. 22, (1988), p. 170–181.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • N. M. Atakishiyev
    • 1
  • S. K. Suslov
    • 2
  1. 1.Physics InstituteBakuAzerbaijan
  2. 2.Kurchatov Institute of Atomic EnergyMoscowRussia

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