Brazilian Journal of Probability and Statistics

Hypergeometric functions where two arguments differ by an integer

Christopher S. Withers and Saralees Nadarajah

Full-text: Open access

Abstract

If $\alpha_{1}-\beta_{1}$ is an integer, then ${}_{p}F_{q}(\boldsymbol{\alpha} ;\boldsymbol{\beta} :z)$ can be expressed in terms of ${}_{p-1}F_{q-1}$. This leads to a conjectured generalization of Kummer’s transformation from ${}_{1}F_{1}$ to ${}_{p}F_{q}$. Applications are given for noncentral chi-square and Student’s $t$ distributions.

Article information

Source
Braz. J. Probab. Stat., Volume 28, Number 1 (2014), 140-149.

Dates
First available in Project Euclid: 5 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1391611342

Digital Object Identifier
doi:10.1214/12-BJPS199

Mathematical Reviews number (MathSciNet)
MR3165433

Zentralblatt MATH identifier
06291465

Keywords
Generalized hypergeometric functions Kummer’s transformation

Citation

Withers, Christopher S.; Nadarajah, Saralees. Hypergeometric functions where two arguments differ by an integer. Braz. J. Probab. Stat. 28 (2014), no. 1, 140--149. doi:10.1214/12-BJPS199. https://projecteuclid.org/euclid.bjps/1391611342


Export citation

References

  • Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions. Applied Mathematics Series 55. U.S. Department of Commerce, National Bureau of Standards.
  • Gradshteyn, I. S. and Ryzhik, I. M. (2007). Tables of Integrals, Series and Products, 7th ed. New York: Academic Press.
  • Johnson, N. L. and Kotz, S. (1970). Continuous Univariate Distributions, Vol. 2. New York: Houghton Miflin.
  • Mathai, A. M. and Saxena, R. K. (1973). Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Lecture Notes in Mathematics 348. Berlin: Springer Verlag.
  • Mathai, A. M. and Saxena, R. K. (1978). The $H$-Function with Applications in Statistics and Other Disciplines. New York: Wiley.
  • Mathai, A. M., Saxena, R. K. and Haubold, H. J. (2010). The $H$-Function: Theory and Applications. New York: Springer Verlag.