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q-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials


We obtain algebraic relations (identities) for q-numbers that do not contain q α-factors. We derive a formula that expresses any q-number [x] in terms of the q-number [2]. We establish the relationship between the q-numbers [n] and the Fibonacci numbers, Chebyshev polynomials, and other special functions. The sums of combinations of q-numbers, in particular, the sums of their powers, are calculated. Linear and bilinear generating functions are found for “natural” q-numbers.

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  1. T. H. Koornwinder, “Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials,” Ned Acad. Wetensch. Proc. Ser. A, 92, No. 2, 97–102 (1989).

    MathSciNet  Google Scholar 

  2. L. C. Biedenharn and M. A. Lohe, Quantum Groups Symmetry and q-Tensor Algebras, World Scientific, Singapore (1995).

    Google Scholar 

  3. Ya. A. Smorodinskii, A. A. Shelepin, and L. A. Shelepin, “Group and probability foundations of quantum theory,” Usp. Fiz. Nauk, 162, No. 3, 2–95 (1992).

    Google Scholar 

  4. G. Gasper and M. Rahman, Basic Hypergeometric Functions, Cambridge University Press, Cambridge (1990).

    Google Scholar 

  5. A. M. Gavrilik and A. U. Klimyk, “Representations of q-deformed algebras Uq(SO2, 1) and Uq(SO3, 1),” J. Math. Phys., 35, No. 10, 3670–3686 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. V. Kuzhel’, Mathematical Improvisations [in Russian], Vyshcha Shkola, Kiev (1983).

    Google Scholar 

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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1055–1063, August, 1998.

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Kachurik, I.I. q-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials. Ukr Math J 50, 1201–1211 (1998).

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  • Orthogonal Polynomial
  • Quantum Group
  • Chebyshev Polynomial
  • Arithmetic Progression
  • Summation Formula