Orthogonal Polynomials with Ultra-Exponential Weight Functions: An Explicit Solution to the Ditkin–Prudnikov Problem


New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations in terms of the Mellin–Barnes and Laplace integrals. A notion of the composition polynomial orthogonality is introduced. The corresponding advantages of this orthogonality to discover new sequences of polynomials and their relations to the corresponding multiple orthogonal polynomial ensembles are shown.

This is a preview of subscription content, access via your institution.


  1. 1.

    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, London (1978)

    MATH  Google Scholar 

  2. 2.

    Coussement, E., Van Assche, W.: Some properties of multiple orthogonal polynomials associated with Macdonald functions. J. Comput. Appl. Math. 133, 253–261 (2001)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Dattoli, G., Ricci, P.E., Khomasuridze, I.: Operational methods, special polynomial and functions and solution of partial differential equations. Integral Transf. Spec. Funct. 15(4), 309–321 (2004)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ditkin, V.A., Prudnikov, A.P.: Integral transforms. Mathematical analysis, 1966, 7-82. Akad. Nauk SSSR Inst. Nauchn. Informacii, Moscow, 1967 (in Russian)

  5. 5.

    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, Vols. I and II. McGraw-Hill, New York, London, Toronto (1953)

  6. 6.

    Ismail, M.E.H.: Bessel functions and the infinite divisibility of the student \(t\)-distribution. Ann. Prob. 5, 582–585 (1977)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kuijlaars, A.B.J., Zhang, L.: Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard eddge scaling limits. Commun. Math. Phys. 332, 759–781 (2014)

    Article  Google Scholar 

  8. 8.

    Loureiro, A.F., Yakubovich, S.: Central factorials under the Kontorovich–Lebedev transform of polynomials. Integral Transf. Spec. Funct. 24(3), 217–238 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Prudnikov, A.P.: Orthogonal polynomials with ultra-exponential weight functions, in: W. Van Assche (Ed.), Open Problems, J. Comput. Appl. Math. 48, 239-241, (1993)

  10. 10.

    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Vol. I: Elementary Functions, Gordon and Breach, New York, London, 1986; Vol. II: Special Functions, Gordon and Breach, New York, London, 1986; Vol. III: More Special Functions, Gordon and Breach, New York, London (1990)

  11. 11.

    Riordan, J.: Combinatorial Identities. Wiley, New York (1968)

    MATH  Google Scholar 

  12. 12.

    Titchmarsh, E.C.: An Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1937)

    MATH  Google Scholar 

  13. 13.

    Van Assche, W., Yakubovich, S.: Multiple orthogonal polynomials associated with Macdonald functions. Integral Transf. Spec. Funct. 9(3), 229–244 (2000)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Viskov, O.V., Srivastava, H.M.: New approaches to certain identities involving differential operators. J. Math. Anal. Appl. 186, 1–10 (1994)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Yakubovich, S., Luchko, Y.: The Hypergeometric Approach to Integral Transforms and Convolutions, Kluwer Academic Publishers. Math. Appl. 287, (1994)

Download references

Author information



Corresponding author

Correspondence to S. Yakubovich.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work was partially supported by CMUP, which is financed by national funds through FCT (Portugal) under the project with reference UIDB/00144/2020. The author thanks Marco Martins Afonso for necessary numerical calculations and verifications of some formulas. Finally, the author is sincerely indebted to referees for useful comments and suggestions which rather improved the presentation of the paper.

Communicated by Erik Koelink.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yakubovich, S. Orthogonal Polynomials with Ultra-Exponential Weight Functions: An Explicit Solution to the Ditkin–Prudnikov Problem. Constr Approx 53, 1–38 (2021). https://doi.org/10.1007/s00365-020-09523-0

Download citation


  • Orthogonal polynomials
  • Modified Bessel functions
  • Meijer G-function
  • Mellin transform
  • Associated Laguerre polynomials
  • Multiple orthogonal polynomials

Mathematics Subject Classification

  • 33C47
  • 33C45
  • 33C10
  • 44A15
  • 42C05