The behavior of the zerobalanced hypergeometric series near the boundary of its convergence region
Authors:
Megumi Saigo and H. M. Srivastava
Journal:
Proc. Amer. Math. Soc. 110 (1990), 7176
MSC:
Primary 33C20
MathSciNet review:
1036991
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Abstract: For a zerobalanced generalized hypergeometric function , the authors prove a formula exhibiting its behavior near the boundary point of the region of convergence of the series defining it. The result established here provides an interesting extension of a formula which appeared in one of Ramanujan's celebrated Notebooks; it also serves to solve the problem posed by R. J. Evans [5].
 [1]
P. Appell et J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite, GauthierVillars, Paris, 1926. MR 0038393 (12:397b)
 [2]
B. C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. London Math. Soc. 15 (1983), 273320. MR 703753 (85a:01043)
 [3]
W. Bühring, The behavior at unit argument of the hypergeometric function , SIAM J. Math. Anal. 18 (1987), 12271234. MR 902328 (88j:33004)
 [4]
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. I, McGrawHill, New York, 1953.
 [5]
R. J. Evans, Ramanujan's second notebook: Asymptotic expansions for hypergeometric series and related functions, in Ramanujan Revisited (G. E. Andrews et al., eds.), (Proc. of the Ramanujan Centenary Conference, Univ. of Illinois, June 15, 1987) Academic Press, New York, 1988, pp. 537560. MR 938978 (89c:33003)
 [6]
R. J. Evans and D. Stanton, Asymptotic formulas for zerobalanced hypergeometric series, SIAM J. Math. Anal. 15 (1984), 10101020. MR 755861 (85i:33003)
 [7]
O. I. Marichev and S. L. Kalla, Behaviour of hypergeometric function in the vicinity of unity, Rev. Técn. Fac. Ingr. Univ. Zulia 7 (1984), 18. MR 781315 (86e:33006)
 [8]
N. E. Nørlund, Hypergeometric functions, Acta Math. 94 (1955), 289349. MR 0074585 (17:610d)
 [9]
S. Ramanujan, Notebooks of Srinivasa Ramanujan, vol. 2, Tata Inst. of Fundamental Research, Bombay, 1957. MR 0099904 (20:6340)
 [10]
M. Saigo, A certain boundary value problem for the EulerDarboux equation. II and III, Math. Japon. 25 (1980), 211220; 26 (1981), 103119. MR 580227 (84a:35234)
 [11]
, On a property of the Appell hypergeometric function , Math. Rep. College General Ed. Kyushu Univ. 12 (1980), 6367.
 [12]
, On properties of the Appell hypergeometric functions and and the generalized Gauss function , Bull. Central Res. Inst. Fukuoka Univ. 66 (1983), 2732.
 [13]
, On properties of the Lauricella hypergeometric function , Bull. Central Res. Inst. Fukuoka Univ. 104 (1988), 1331.
 [14]
, On properties of hypergeometric functions of three variables, and , Rend. Circ. Mat. Palermo (2) 37 (1988), 449468.
 [15]
M. Saigo and H. M. Srivastava, The behaviors of the Appell double hypergeometric series and certain Lauricella triple hypergeometric series near the boundaries of their convergence regions, Fukuoka Univ. Sci. Rep. 19 (1989), 110. MR 996550 (90c:33007)
 [16]
, The behavior of the Lauricella hypergeometric series in variables near the boundaries of its convergence region, Univ. of Victoria Report No. DM480IR, 1988, pp. 123.
 [17]
H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, Wiley, New York, 1985. MR 834385 (87f:33015)
 [18]
H. M. Srivastava and M. Saigo, Multiplication of fractional calculus operators and boundary value problems involving the EulerDarboux equation, J. Math. Anal. Appl. 121 (1987), 325369. MR 872230 (88c:26008)
 [1]
 P. Appell et J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite, GauthierVillars, Paris, 1926. MR 0038393 (12:397b)
 [2]
 B. C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. London Math. Soc. 15 (1983), 273320. MR 703753 (85a:01043)
 [3]
 W. Bühring, The behavior at unit argument of the hypergeometric function , SIAM J. Math. Anal. 18 (1987), 12271234. MR 902328 (88j:33004)
 [4]
 A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. I, McGrawHill, New York, 1953.
 [5]
 R. J. Evans, Ramanujan's second notebook: Asymptotic expansions for hypergeometric series and related functions, in Ramanujan Revisited (G. E. Andrews et al., eds.), (Proc. of the Ramanujan Centenary Conference, Univ. of Illinois, June 15, 1987) Academic Press, New York, 1988, pp. 537560. MR 938978 (89c:33003)
 [6]
 R. J. Evans and D. Stanton, Asymptotic formulas for zerobalanced hypergeometric series, SIAM J. Math. Anal. 15 (1984), 10101020. MR 755861 (85i:33003)
 [7]
 O. I. Marichev and S. L. Kalla, Behaviour of hypergeometric function in the vicinity of unity, Rev. Técn. Fac. Ingr. Univ. Zulia 7 (1984), 18. MR 781315 (86e:33006)
 [8]
 N. E. Nørlund, Hypergeometric functions, Acta Math. 94 (1955), 289349. MR 0074585 (17:610d)
 [9]
 S. Ramanujan, Notebooks of Srinivasa Ramanujan, vol. 2, Tata Inst. of Fundamental Research, Bombay, 1957. MR 0099904 (20:6340)
 [10]
 M. Saigo, A certain boundary value problem for the EulerDarboux equation. II and III, Math. Japon. 25 (1980), 211220; 26 (1981), 103119. MR 580227 (84a:35234)
 [11]
 , On a property of the Appell hypergeometric function , Math. Rep. College General Ed. Kyushu Univ. 12 (1980), 6367.
 [12]
 , On properties of the Appell hypergeometric functions and and the generalized Gauss function , Bull. Central Res. Inst. Fukuoka Univ. 66 (1983), 2732.
 [13]
 , On properties of the Lauricella hypergeometric function , Bull. Central Res. Inst. Fukuoka Univ. 104 (1988), 1331.
 [14]
 , On properties of hypergeometric functions of three variables, and , Rend. Circ. Mat. Palermo (2) 37 (1988), 449468.
 [15]
 M. Saigo and H. M. Srivastava, The behaviors of the Appell double hypergeometric series and certain Lauricella triple hypergeometric series near the boundaries of their convergence regions, Fukuoka Univ. Sci. Rep. 19 (1989), 110. MR 996550 (90c:33007)
 [16]
 , The behavior of the Lauricella hypergeometric series in variables near the boundaries of its convergence region, Univ. of Victoria Report No. DM480IR, 1988, pp. 123.
 [17]
 H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, Wiley, New York, 1985. MR 834385 (87f:33015)
 [18]
 H. M. Srivastava and M. Saigo, Multiplication of fractional calculus operators and boundary value problems involving the EulerDarboux equation, J. Math. Anal. Appl. 121 (1987), 325369. MR 872230 (88c:26008)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010369911
PII:
S 00029939(1990)10369911
Keywords:
Hypergeometric functions,
boundary value problems,
EulerDarboux equation,
analytic continuation,
asymptotic formula,
Kampé de Fériet series
Article copyright:
© Copyright 1990
American Mathematical Society
