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On Quadruple q-Hypergeometric Functions and Diverse Generalizations to n Variables in the Spirit of Exton

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Abstract

The purpose of this article is to study convergence regions and q-integral representations of certain non-symmetric q-Lauricella functions and quadruple functions in the spirit of Exton. In the process, we slightly improve Exton’s original formulas, notation, and convergence regions. There are three so-called q-real numbers, which are briefly introduced. These numbers occur both in the q-integrals and in the convergence regions. When q-integral expressions for the symmetric \({\Phi _{A }^{(n)}}\) and \({\Phi _{D }^{(n)}}\) are used in the proofs, in the first case, third q-real numbers occur in the q-integrals. When \({\Phi _{D }^{(n)}}\) is used in the proofs, the formulas are simpler, because the latter function has greater convergence region. Similarly, multiple q-Horn functions are briefly discussed.

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References

  1. Chandel Singh, R.C.: On some Multiple hypergeometric functions related to Lauricella’s functions. Jñānābha 3, 119–136 (1973)

  2. Ernst, T.: Convergence aspects for \(q\)-Lauricella functions 1 Adv. Stud. Contemp. Math. 22(1), 35–50 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Ernst, T.: A Comprehensive Treatment of \(q\)-Calculus. Birkhäuser, Basel (2012)

    Book  MATH  Google Scholar 

  4. Ernst, T.: On the \(q\)-analogues of Srivastava’s triple hypergeometric functions. Axioms 2(2), 85–99 (2013)

    Article  MATH  Google Scholar 

  5. Ernst, T.: On the symmetric \(q\)-Lauricella functions. Proc. Jangjeon Math. Soc. 19(2), 319–344 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Ernst, T.: On Eulerian \(q\)-integrals for single and multiple \(q\)-hypergeometric series. Commun. Korean Math. Soc. 33(1), 179–196 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Ernst, T.: On the complex \(q\)-Appell polynomials. Ann. Univ. Mariae Curie-Sklodowska Sect. A 74(1), 31–43 (2020)

    MathSciNet  MATH  Google Scholar 

  8. Ernst, T.: On the triple Lauricella–Horn–Karlsson \(q\)-hypergeometric functions. Axioms 9(3) (2020)

  9. Ernst, T.: On confluent \(q\)-hypergeometric functions. In: Proceedings Srivastava 80 Years, Applied Analysis and Optimization, vol. 4, no. 3, pp. 385–404 (2020)

  10. Ernst, T.: Three algebraic number systems based on the \(q\)-addition with applications. Ann. Univ. Marie Curie Sect. A 75(2), 45–71 (2021)

    MathSciNet  MATH  Google Scholar 

  11. Ernst, T.: Aspects opérateurs, systèmes d’équations aux \(q\)-différences et \(q\)-intégrales pour différents types de \(q\)-fonctions multiples dans l’esprit de Horn, Debiard et Gaveau, Operator aspects, systems of \(q\)-difference equations et \(q\)-integrals for different types of multiple \(q\)-functions in the spirit of Horn, Debiard and Gaveau (French). Algebras Groups Geom. (2023) (accepted)

  12. Exton, H.: On certain confluent hypergeometric functions of three variables. Ganita 21(2), 79–92 (1970)

    MathSciNet  MATH  Google Scholar 

  13. Exton, H.: Certain hypergeometric functions of four variables. Bull. Soc. Math. Greece (N.S.) 13(1–2), 104–113 (1972)

    MathSciNet  MATH  Google Scholar 

  14. Exton, H.: Multiple Hypergeometric Functions and Applications. Mathematics and Its Applications. Ellis Horwood Ltd./Halsted Press [Wiley], Chichester/New York (1976)

  15. Karlsson, P.: On intermediate Lauricella functions. Jnanabha 16, 211–222 (1986)

    MathSciNet  MATH  Google Scholar 

  16. Lauricella, G.: On hypergeometric functions of several variables. (Sulle Funzioni Ipergeometriche a piu Variabili.) (Italian) Rend. Circ. Mat. Palermo 7, 111–158 (1893)

  17. Qureshi, M.I., Quraishi, K.A., Khan, B., Arora, A.: Transformations associated with quadruple hypergeometric functions of Exton and Srivastava. Asia Pac. J. Math. 4(1), 38–48 (2017)

    MATH  Google Scholar 

  18. Srivastava, H.M.: A formal extension of certain generating functions. Glas. Mat. III. Ser. 5(25), 229–239 (1970)

    MathSciNet  MATH  Google Scholar 

  19. Srivastava, H.M.: A formal extension of certain generating functions II. Glas. Mat. III. Ser. 6(26), 35–44 (1971)

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Acknowledgements

We thank Harold Exton for his most interesting articles and books on multiple hypergeometric functions. Thanks to Karl-Heinz Fieseler and Christer Kiselman for their generous help on improving the text.

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  1. Department of Mathematics, Uppsala University, P.O. Box 480, 751 06, Uppsala, Sweden

    Thomas Ernst

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Correspondence to Thomas Ernst.

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Communicated by Saeid Maghsoudi.

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Ernst, T. On Quadruple q-Hypergeometric Functions and Diverse Generalizations to n Variables in the Spirit of Exton. Bull. Iran. Math. Soc. 49, 73 (2023). https://doi.org/10.1007/s41980-023-00809-0

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