Year 2013, Volume 42, Issue 3, Pages 281 - 287 2013-03-01

Some Summation Formulas for the Hypergeometric Series r+2Fr+1(½)

Y. S. Kim [1] , A. K. Rathie [2] , U. P [3] , - - [4] , R. B. Paris [5]

256 520

The aim of this paper is to obtain explicit expressions of the generalizedhypergeometric functionr+2 F r+1 a, b, (a + b + j + 1),(f r + m r ) (f r ) ; for j = 0, ±1, . . . , ±5, where r pairs of numeratorial and denominatorialparameters differ by positive integers mr . The results are derived withthe help of an expansion in terms of a finite sum ofF ( ) functions anda generalization of Gauss’ second summation theorem due to Lavoie etal. [J. Comput. Appl. Math. 72, 293–300 (1996)]. Some special andlimiting cases are also given.
Generalized hypergeometric series, Generalized Gauss summation theorem
  • Abramowitz, M. and Stegun, I. A. (Eds.) Handbook of Mathematical Functions (Dover, New York, 1965).
  • Karlsson, P. W. Hypergeometric functions with integral parameter differences, J. Math. Phys. 12, 270–271, 1971.
  • Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Watson’s theorem on the sum of a 3 F 2 , Indian J. Math. 34, 23–32, 1992.
  • Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Dixon’s theorem on the sum of a 3 F 2 , Math. Comp. 62, 267–276, 1994.
  • Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Whipple’s theorem on the sum of a 3 F 2 , J. Comput. Appl. Math. 72, 293–300, 1996.
  • Luke, Y. L. Mathematical Functions and Their Approximations (Academic Press, New York, 1975).
  • Miller, A. R. Certain summation and transformation formulas for generalized hypergeometric series, J. Comput. Appl. Math. 231, 964–972, 2009.
  • Miller, A. R. and Paris, R. B. Certain transformations and summations for the generalized hypergeometric series with integral parameter differences, Integral Transforms and Special Functions, 22, 67–77, 2011.
  • Miller, A. R. and Paris, R. B. Euler-type transformations for the generalized hypergeometric function r+2 F r+1 (x), Zeitschrift angew. Math. Phys., 62, 31–45, 2011.
  • Miller, A. R. and Paris, R. B. On a result related to transformations and summations of generalized hypergeometric series, Math. Communications, 17, 205–210, 2012.
  • Miller, A. R. and Paris, R. B. Transformation formulas for the generalized hypergeometric function with integral parameter differences, Rocky Mountain J. Math., 43, 291-327, 2013. Miller, A. R. and Srivastava, H. M. Karlsson–Minton summation theorems for the generalized hypergeometric series of unit argument, Integral Transforms and Special Functions, 21, 603–612, 2010.
  • Minton, B. M. Generalized hypergeometric function of unit argument, J. Math. Phys., 11, 1375–1376, 1970.
  • Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. Integrals and Series: More Special Functions, vol. 3 (Gordon and Breach, New York, 1990).
  • Rathie, A. K. and Pog´ any, T. K. New summation formula for 3 F 2 ( 1 2 ) and a Kummer-type II transformation of 2 F 2 (x), Mathematical Communications, 13, 63–66, 2008.
  • Slater, L. J. Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966).
Primary Language tr
Journal Section Mathematics
Authors

Author: Y. S. Kim

Author: A. K. Rathie

Author: U. P

Author: - -

Author: R. B. Paris

Dates

Publication Date: March 1, 2013

Bibtex @ { hujms101280, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {Hacettepe University}, year = {2013}, volume = {42}, pages = {281 - 287}, doi = {}, title = {Some Summation Formulas for the Hypergeometric Series r+2Fr+1(\½)}, key = {cite}, author = {Kim, Y. S. and Rathie, A. K. and P, U. and -, - and Paris, R. B.} }
APA Kim, Y , Rathie, A , P, U , -, - , Paris, R . (2013). Some Summation Formulas for the Hypergeometric Series r+2Fr+1(½). Hacettepe Journal of Mathematics and Statistics, 42 (3), 281-287. Retrieved from http://dergipark.org.tr/hujms/issue/7748/101280
MLA Kim, Y , Rathie, A , P, U , -, - , Paris, R . "Some Summation Formulas for the Hypergeometric Series r+2Fr+1(&frac12;)". Hacettepe Journal of Mathematics and Statistics 42 (2013): 281-287 <http://dergipark.org.tr/hujms/issue/7748/101280>
Chicago Kim, Y , Rathie, A , P, U , -, - , Paris, R . "Some Summation Formulas for the Hypergeometric Series r+2Fr+1(&frac12;)". Hacettepe Journal of Mathematics and Statistics 42 (2013): 281-287
RIS TY - JOUR T1 - Some Summation Formulas for the Hypergeometric Series r+2Fr+1(&frac12;) AU - Y. S. Kim , A. K. Rathie , U. P , - - , R. B. Paris Y1 - 2013 PY - 2013 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 281 EP - 287 VL - 42 IS - 3 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2019 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Some Summation Formulas for the Hypergeometric Series r+2Fr+1(&frac12;) %A Y. S. Kim , A. K. Rathie , U. P , - - , R. B. Paris %T Some Summation Formulas for the Hypergeometric Series r+2Fr+1(&frac12;) %D 2013 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 42 %N 3 %R %U
ISNAD Kim, Y. S. , Rathie, A. K. , P, U. , -, - , Paris, R. B. . "Some Summation Formulas for the Hypergeometric Series r+2Fr+1(&frac12;)". Hacettepe Journal of Mathematics and Statistics 42 / 3 (March 2013): 281-287.
AMA Kim Y , Rathie A , P U , - - , Paris R . Some Summation Formulas for the Hypergeometric Series r+2Fr+1(&frac12;). Hacettepe Journal of Mathematics and Statistics. 2013; 42(3): 281-287.
Vancouver Kim Y , Rathie A , P U , - - , Paris R . Some Summation Formulas for the Hypergeometric Series r+2Fr+1(&frac12;). Hacettepe Journal of Mathematics and Statistics. 2013; 42(3): 287-281.