About the Project

Ramanujan beta integral

AdvancedHelp

(0.003 seconds)

8 matching pages

1: 5.13 Integrals
Ramanujan’s Beta Integral
2: Bibliography P
  • P. I. Pastro (1985) Orthogonal polynomials and some q -beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.
  • 3: 17.13 Integrals
    §17.13 Integrals
    17.13.2 c d ( q x / c ; q ) ( q x / d ; q ) ( x q α / c ; q ) ( x q β / d ; q ) d q x = Γ q ( α ) Γ q ( β ) Γ q ( α + β ) c d c + d ( c / d ; q ) ( d / c ; q ) ( q β c / d ; q ) ( q α d / c ; q ) .
    Ramanujan’s Integrals
    17.13.3 0 t α 1 ( t q α + β ; q ) ( t ; q ) d t = Γ ( α ) Γ ( 1 α ) Γ q ( β ) Γ q ( 1 α ) Γ q ( α + β ) ,
    Askey (1980) conjectured extensions of the foregoing integrals that are closely related to Macdonald (1982). …
    4: Bibliography R
  • S. Ramanujan (1921) Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.
  • S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,
  • S. Ramanujan (1962) Collected Papers of Srinivasa Ramanujan. Chelsea Publishing Co., New York.
  • Ju. M. Rappoport (1979) Tablitsy modifitsirovannykh funktsii Besselya K 1 2 + i β ( x ) . “Nauka”, Moscow (Russian).
  • G. B. Rybicki (1989) Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.
  • 5: Bibliography C
  • M. Carmignani and A. Tortorici Macaluso (1985) Calcolo delle funzioni speciali Γ ( x ) , log Γ ( x ) , β ( x , y ) , erf ( x ) , erfc ( x ) alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).
  • H. H. Chan (1998) On Ramanujan’s cubic transformation formula for F 1 2 ( 1 3 , 2 3 ; 1 ; z ) . Math. Proc. Cambridge Philos. Soc. 124 (2), pp. 193–204.
  • R. Chattamvelli and R. Shanmugam (1997) Algorithm AS 310. Computing the non-central beta distribution function. Appl. Statist. 46 (1), pp. 146–156.
  • D. V. Chudnovsky and G. V. Chudnovsky (1988) Approximations and Complex Multiplication According to Ramanujan. In Ramanujan Revisited (Urbana-Champaign, Ill., 1987), G. E. Andrews, R. A. Askey, B. C. Bernd, K. G. Ramanathan, and R. A. Rankin (Eds.), pp. 375–472.
  • D. Cvijović and J. Klinowski (1999) Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.
  • 6: Bibliography K
  • J. Kamimoto (1998) On an integral of Hardy and Littlewood. Kyushu J. Math. 52 (1), pp. 249–263.
  • M. Katsurada (2003) Asymptotic expansions of certain q -series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.
  • D. A. Kofke (2004) Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)]. J. Chem. Phys. 121 (2), pp. 1167.
  • T. H. Koornwinder (1984b) Orthogonal polynomials with weight function ( 1 x ) α ( 1 + x ) β + M δ ( x + 1 ) + N δ ( x 1 ) . Canad. Math. Bull. 27 (2), pp. 205–214.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
  • 7: Bibliography D
  • B. Davies (1984) Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.
  • L. Debnath and D. Bhatta (2015) Integral transforms and their applications. Third edition, CRC Press, Boca Raton, FL.
  • A. R. DiDonato and A. H. Morris (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.
  • H. Ding, K. I. Gross, and D. St. P. Richards (1996) Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.
  • J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.
  • 8: Bibliography
  • C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson (1985) Chapter 16 of Ramanujan’s second notebook: Theta-functions and q -series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.
  • W. A. Al-Salam and M. E. H. Ismail (1994) A q -beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.
  • G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π , and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.
  • G. E. Andrews, R. A. Askey, B. C. Berndt, and R. A. Rankin (Eds.) (1988) Ramanujan Revisited. Academic Press Inc., Boston, MA.
  • G. E. Andrews (1984) Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.