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Dergiler / Turkish Journal of Mathematics / 2020 / Cilt: 44 - Sayı: 2

On extensions of two results due to Ramanujan

Yong Sup KIM
Arjunkumar RATHIE
Rıchard B. PARIS
Sayfa
348–355
Yayın Tarihi
2020-03-01
DOI
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In addition we generalize a summation formula to include integral parameter differences.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:The aim in this note is to provide a generalization of an interesting entry in Ramanujan's notebooks that relate sums involving the derivatives of a function $\varphi\left t\right $ evaluated at 0 and 1.
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Temel Bulgular & Veri:The generalization obtained is derived with the help of expressions for the sum of terminating ${}_3F_2$ hypergeometric functions of argument equal to 2, recently obtained by Kim et. al. [Two results for the terminating ${}_3F_2$ 2 with applications, Bulletin of the Korean Mathematical Society 2012; 49: 621-633].
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Sonuç & Katkı:In addition we generalize a summation formula to include integral parameter differences.
🔬Tür:Araştırma Makalesi
📍Kapsam:2012 Verileri
⚙️Değişkenler:Hypergeometric series · Ramanujan's sum · sums of Hermite polynomials

Abstract

The aim in this note is to provide a generalization of an interesting entry in Ramanujan's notebooks that relate sums involving the derivatives of a function $\varphi\left t\right $ evaluated at 0 and 1. The generalization obtained is derived with the help of expressions for the sum of terminating ${}_3F_2$ hypergeometric functions of argument equal to 2, recently obtained by Kim et. al. [Two results for the terminating ${}_3F_2$ 2 with applications, Bulletin of the Korean Mathematical Society 2012; 49: 621-633]. Several special cases are given. In addition we generalize a summation formula to include integral parameter differences.

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Bu Makaleye Atıf Yap

KIM, Y. S., RATHIE, A., PARIS, R. B. (2020). On extensions of two results due to Ramanujan. Turkish Journal of Mathematics, Cilt: 44 - Sayı: 2, 348–355. https://ajindex.com/articles/185518

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Kaynakça (10)

  1. Berndt BC. Ramanujan’s Notebooks. Part II. New York, NY, USA: Springer, 1987
  2. Choi J, Rathie AK. Two formulas contiguous to a quadratic transformation due to Kummer with an application. Hacettepe Journal of Mathematics 2011; 40 (6): 885-894
  3. Kim YS, Choi J, Rathie AK. Two results for the terminating 3F2(2) with applications. Bulletin of Korean Mathematics Society 2012; 49 (3): 621-633
  4. Kim YS, Rathie AK, Paris RB. Generalization of two theorems due to Ramanujan. Integral Transforms and Special Functions 2012; 24: 314-323
  5. Miller AR, Paris RB. On a result related to transformations and summations of generalized hypergeometric series. Mathematical Communications 2012; 17: 205-210
  6. Miller AR, Paris RB. Transformation formulas for the generalized hypergeometric function with integral parameter differences. Rocky Mountain Journal of Mathematics 2013; 43: 291-327
  7. Miller AR, Srivastava HM. Karlsson–Minton summation theorems for the generalized hypergeometric series of unit argument. Integral Transforms and Special Functions 2010; 21: 603-612
  8. Olver FWJ, Lozier DW, Boisvert RF, Clark CW (editors). NIST Handbook of Mathematical Functions. Cambridge, UK: Cambridge University Press, 2010
  9. Rathie AK, Pogány TK. New summation formula for 3F2( 1 2 ) and a Kummer-type II transformation of 2F2(x). Mathematical Communications 2008; 13: 63-66
  10. Slater LJ. Generalized Hypergeometric Functions. Cambridge, UK: Cambridge University Press, 1966.