On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton
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Thomas Ernst
Abstract
The purpose of this article is to find many solutions of the canonical q-difference equations for q-Appell functions Φ2, Φ3, Φ4, starting with the first four ones, previously published. Furthermore, q-Laplace integral expressions for two of these functions in the form q-confluent functions are used to find more solutions. In order to explain the discrepancies between meromorphic continuations of multiple q-hypergeometric functions and q-analogues of analytic continuations of corresponding multiple hypergeometric functions, the notion pseudo-meromorphic continuation is introduced to denote equivalence classes of solutions of other q-analogues of systems of partial differential equations. To this end, we introduce equivalence classes of (systems of) partial q-difference equations. Some q-integrals for corresponding q-Appell functions and the second q-Horn function are also proved. In the process, we find a meromorphic continuation of the second q-Horn function. In some formulas, abbreviated notations are used together with combinatorial reasonings. Many proofs use formulas from our first book, which are quoted in detail.
(Communicated by Giuseppe Di Fazio)
References
[1] Appell, P.—Kampé De Fériet, J.: Fonctions Hypergéométriques et Hypersphériques, Paris, 1926.Suche in Google Scholar
[2] Borngässer, L.: Über hypergeometrische Funktionen zweier Veränderlichen, (German) Darmstadt: Diss. 1933.Suche in Google Scholar
[3] Debiard, A.—Gaveau, B.: Hypergeometric Symbolic Calculus I: Systems of two symbolic hypergeometric equations, Bull. Sci. Math. 126(10) (2002), 773–829.10.1016/S0007-4497(02)01143-0Suche in Google Scholar
[4] Ernst, T.: Multiple q-hypergeometric transformations involving q-integrals, Proceedings of the 9th Annual Conference of the Society for Special Functions and their Applications (SSFA) 9 (2011), 91-99.Suche in Google Scholar
[5] Ernst, T.: A Comprehensive Treatment of q-Calculus, Birkhäuser, 2012.10.1007/978-3-0348-0431-8Suche in Google Scholar
[6] Ernst, T.: Convergence aspects for q-Lauricella functions 1, Adv. Studies Contemp. Math. 22(1) (2012), 35–50.Suche in Google Scholar
[7] Ernst, T.: Convergence aspects of for q-Appell functions I, J. Indian Math. Soc. 81(1–2) (2014), 67–77.Suche in Google Scholar
[8] Ernst, T.: On Eulerian q-integrals for single and multiple q-hypergeometric series, Commun. Korean Math. Soc. 33(1) (2018), 179–196.Suche in Google Scholar
[9] Ernst, T.: On confluent q-hypergeometric functions, Proceedings Srivastava 80 years, Applied Analysis and Optimization Appl. Anal. Optim. 4, (3)(2020) , 385–404.Suche in Google Scholar
[10] Ernst, T.: On the triple LauricellaHornKarlsson q-hypergeometric functions, Axioms 9(3) (2020).10.3390/axioms9030093Suche in Google Scholar
[11] Ernst, T.: On various formulas with q-integrals and their applications to q-hypergeometric functions, European J. Pure and Applied Mathematics 13(5) (2020), 1241–1259.10.29020/nybg.ejpam.v13i5.3755Suche in Google Scholar
[12] Ernst, T.: Three algebraic number systems based on the q-addition with applications, Ann. Univ. Marie Curie, Sect. A 75 (2) (2021), 45–71.10.17951/a.2021.75.2.45-71Suche in Google Scholar
[13] Ernst, T.: A New q-hypergeometric Symbolic Calculus in the Spirit of Horn, Borngässer, Debiard and Gaveau, Axioms 11 (2) (2022), artikel-id 6410.3390/axioms11020064Suche in Google Scholar
[14] Ernst, T.: Meromorphic continuations of multiple q-hypergeometric functions, Proc. Jangjeon Math. Soc. 26, 4 (2023), 443–454.Suche in Google Scholar
[15] 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 differential equations et q-integrals for different types of multiple q-functions in the spirit of Horn, Debiard and Gaveau (French). Algebras Groups Geom. (2023) 39.10.29083/AGG.39.01.2023/SC1Suche in Google Scholar
[16] Exton, H.: Multiple hypergeometric functions and applications, Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney 1976.Suche in Google Scholar
[17] Exton, H.: Handbook of hypergeometric integrals. Theory, applications, tables, computer programs. Chichester; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney 1978.Suche in Google Scholar
[18] Exton, H.: On the system of partial differential equations associated with Appells function F4, J. Phys. A, Math. Gen. 28(3) (1995), 631–641.10.1088/0305-4470/28/3/017Suche in Google Scholar
[19] Gasper, G.—Rahman, M.: Basic Hypergeometric Series, Cambridge, 1990.Suche in Google Scholar
[20] Horn, J.: Hypergeometric functions of two variables, (Hypergeometrische Funktionen zweier Veränderlichen). (German) Math. Ann. 105 (1931), 381–407.10.1007/BF01455825Suche in Google Scholar
[21] Mellin, Hj.: Über den Zusammenhang zwischen den linearen Differential- und Differenzengleichungen Acta Math. 25 (1902), 139–164.10.1007/BF02419024Suche in Google Scholar
[22] Mimachi, K.: Connection problem in holonomic q-difference system associated with a Jackson integral of Jordan-Pochhammer type Nagoya Math. J. 116 (1989), 149–161.10.1017/S0027763000001744Suche in Google Scholar
[23] Olsson, P. O. M.: On the integration of the differential equations of five-parametric double-hypergeometric functions of second order, J. Math. Phys. 18 (1977), 1285–1294.10.1063/1.523405Suche in Google Scholar
[24] Srivastava, H. M.—Karlsson, P. W.: Multiple Gaussian hypergeometric series, Ellis Horwood, New York, 1985.Suche in Google Scholar
[25] Watson, G. N.: The continuations of functions defined by generalised hypergeometric series (x) Cambr. Phil. Soc. Trans. 21 (1910), 281–299.Suche in Google Scholar
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Artikel in diesem Heft
- Joins of normal matrices, their spectrum, and applications
- On isbell’s density theorem for bitopological pointfree spaces II
- Every positive integer is a sum of at most n + 2 centered n-gonal numbers
- A generalisation of q-additive functions
- Generalised class groups in dihedral and fake ℤp-extensions
- Singular value bounds with applications to norm and numerical radius inequalities
- Coefficient bounds for convex functions associated with cosine function
- On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton
- Numerical implementation of solving a boundary value problem including both delay and parameter
- Solutions of second order iterative boundary value problems with nonhomogeneous boundary conditions
- Global attractivity of nonlinear delay dynamic equations on time scales via Lyapunov functional method
- On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems
- A note on derivations into annihilators of the ideals of banach algebras
- Characterization of nonlinear mixed skew lie and jordan n-Type derivation on ∗-Algebras
- Linear and uniformly continuous surjections between Cp-spaces over metrizable spaces
- A goodness-of-fit test for testing exponentiality based on normalized dynamic survival extropy
- Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function
