Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T01:17:22.931Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Integral Equations Involving Hypergeometric Functions

Published online by Cambridge University Press:  20 January 2009

E. R. Love
E. R. Love
Affiliation:
University of Edinburgh and University of Melbourne
Rights & Permissions [Opens in a new window]

Summary

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An integral equation of the first kind, with kernel involving a hypergeometric function, is discussed. Conditions sufficient for uniqueness of solutions are given, then conditions necessary for existence of solutions. Conditions sufficient for existence of solutions, only a little stricter than the necessary conditions, are given; and with them two distinct forms of explicit solution. These two forms are associated at first with different ranges of the parameters, but their validity in the complementary ranges is also discussed. Before giving the existence theory a digression is made on a subsidiary integral equation.

Corresponding theorems for another integral equation resembling the main one are deduced from some of the previous theorems. Two more equations of similar form, less closely related, will be considered in another paper. Special cases of some of these four integral equations have been considered recently by Erdélyi, Higgins, Wimp and others.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 15 , Issue 3 , June 1967 , pp. 169 - 198
Copyright
Copyright © Edinburgh Mathematical Society 1967

References

REFERENCES

(1) Li, Ta, A new class of integral transforms, Proc. Amer. Math. Soc. 11 (1960), 290–298.CrossRefGoogle Scholar
(2) Buschman, R. G., An inversion integral, Proc. Amer. Math. Soc. 13 (1962), 675–677.CrossRefGoogle Scholar
(3) Higgins, T. P., An inversion integral for a Gegenbauer transformation, J. Soc. Indust. Appl. Math. 11 (1963), 886–893.CrossRefGoogle Scholar
(4) Srivastava, K. N., A class of integral equations involving ultraspherical polynomials as kernel, Proc. Amer. Math. Soc. 14 (1963), 932–940.CrossRefGoogle Scholar
(5) Erdélyi, A., An integral equation involving Legendre functions, J. Soc. Indust. Appl. Math. 12 (1964), 15–30.CrossRefGoogle Scholar
(6) Higgins, T. P., A hypergeometric function transform, J. Soc. Indust. Appl. Math. 12 (1964), 601–612.CrossRefGoogle Scholar
(7) Wimp, JET, TWO integral transform pairs involving hypergeometric functions, Proc. Glasgow Math. Assoc. 7 (1965), 42–44.CrossRefGoogle Scholar
(8) Srivastava, K. N., Fractional integration and integral equations with polynomial kernels, J. London Math. Soc. 40 (1965), 435–440.CrossRefGoogle Scholar
(9) Kober, H., On fractional integrals and derivatives, Quart. J. Math. (Oxford) 11 (1940), 193–211 (Thm. 2(3.4) (i) with p = 1).CrossRefGoogle Scholar
(10) Erdélyi, A. et al. , Higher Transcendental Functions, vol. 1 (McGraw Hill, 1953).Google Scholar