Araştırma Makalesi
Yıl 2020,
Cilt: 49 Sayı: 5, 1611 - 1624, 06.10.2020
Öz
Kaynakça
- [1] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.
- [2] W.N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J. Math. 7, 105–115, 1936.
- [3] T.J.I’A. Bromwich, An Introduction to the Theory of Infinite Series (2nd ed), Macmillan, New York, 1959.
- [4] X.J. Chen and W. Chu, Closed formulae for a class of terminating $_3F_2(4)$-series, Integral Transforms Spec. Funct. 28 (11), 825–837, 2017.
- [5] X.J. Chen and W. Chu, Terminating $_3F_2(4)$-series extended with three integer parameters, J. Difference Equ. Appl. 24 (8), 1346–1367, 2018.
- [6] W. Chu, Abel’s lemma on summation by parts and Ramanujan’s $_1\psi_1$-series Identity, Aequationes Math. 72 (1-2), 172–176, 2006.
- [7] W. Chu, Abel’s method on summation by parts and hypergeometric series, J. Difference Equ. Appl. 12 (8), 783–798, 2006.
- [8] W. Chu, Bailey’s very well–poised ${_6\psi_6}$-series identity, J. Combin. Theory Ser. 113 (6), 966–979, 2006.
- [9] W. Chu, Abel’s lemma on summation by parts and basic hypergeometric series, Adv. Appl. Math. 39 (4), 490–514, 2007.
- [10] W. Chu, Asymptotic method for Dougall’s bilateral hypergeometric sums, Bull. Sci. Math. 131 (5), 457–468, 2007.
- [11] W. Chu, q-extensions of Dougall’s bilateral ${_2H_2}$-series, Ramanujan J. 25 (1), 121–139, 2011.
- [12] W. Chu, Evaluation of nonterminating hypergeometric $_3F_2(\frac34)$-series, J. Math. Anal. Appl. 450 (1), 490–503, 2017.
- [13] W. Chu and X. Wang, The modified Abel lemma on summation by parts and terminating hypergeometric series identities, Integral Transforms Spec. Funct. 20 (2), 93–118, 2009.
- [14] W. Chu, X. Wang, and D.Y. Zheng, Application of the residue theorem to bilateral hypergeometric series, Matematiche 62 (2), 127–146, 2007.
- [15] A.C. Dixon, Summation of a certain series, Proc. London Math. Soc. 35 (1), 284–291, 1903.
- [16] M.A. Dougall, On Vandermonde’s theorem and some more general expansion, Proc. Edin. Math. Soc. 25, 114–132, 2007.
- [17] R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley Publ. Company, Reading, Massachusetts, 1989.
- [18] M. Jackson, A note on the sum of a particular well–poised $_6H_6$ with argument −1, J. London Math. Soc. 27, 124–126, 1952.
- [19] K. Knopp, Theory and Applications of Infinite Series, Hafner Publishing Company, New York, 1971.
- [20] E.D. Rainville, Special Functions, New York, The Macmillan Company, 1960.
- [21] H.M. Srivastava, Y. Vyas, and K. Fatawat, Extensions of the classical theorems for very well–poised hypergeometric functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113 (367), 2019, https://doi.org/10.1007/s13398-017-0485-5.
- [22] K.R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth, INC. Belmont, California, 1981.
- [23] C. Wang and X. Chen, A short proof for Gosper’s $_7F_6$-series conjecture, J. Math. Anal. Appl. 422 (2), 819–824, 2015.
- [24] C.Wang, J. Dai, and I. Mezo, A nonterminating $_7F_6$-series evaluation, Integral Transforms Spec. Funct. 29 (9), 719–724, 2018.
- [25] F.J.W. Whipple, On well–poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. 24 (2), 247–263, 1926.
Yıl 2020,
Cilt: 49 Sayı: 5, 1611 - 1624, 06.10.2020
Öz
Two classes of nonterminating well--poised series are examined by means of the modified Abel lemma on summation by parts, that leads to several summation and transformation formulae.
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Anahtar Kelimeler
Classical hypergeometric series Abel's lemma on summation by parts Pfaff–Saalschütz theorem Dougall's formulae for $_2H_2$-series and $_5H_5$-series
Kaynakça
- [1] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.
- [2] W.N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J. Math. 7, 105–115, 1936.
- [3] T.J.I’A. Bromwich, An Introduction to the Theory of Infinite Series (2nd ed), Macmillan, New York, 1959.
- [4] X.J. Chen and W. Chu, Closed formulae for a class of terminating $_3F_2(4)$-series, Integral Transforms Spec. Funct. 28 (11), 825–837, 2017.
- [5] X.J. Chen and W. Chu, Terminating $_3F_2(4)$-series extended with three integer parameters, J. Difference Equ. Appl. 24 (8), 1346–1367, 2018.
- [6] W. Chu, Abel’s lemma on summation by parts and Ramanujan’s $_1\psi_1$-series Identity, Aequationes Math. 72 (1-2), 172–176, 2006.
- [7] W. Chu, Abel’s method on summation by parts and hypergeometric series, J. Difference Equ. Appl. 12 (8), 783–798, 2006.
- [8] W. Chu, Bailey’s very well–poised ${_6\psi_6}$-series identity, J. Combin. Theory Ser. 113 (6), 966–979, 2006.
- [9] W. Chu, Abel’s lemma on summation by parts and basic hypergeometric series, Adv. Appl. Math. 39 (4), 490–514, 2007.
- [10] W. Chu, Asymptotic method for Dougall’s bilateral hypergeometric sums, Bull. Sci. Math. 131 (5), 457–468, 2007.
- [11] W. Chu, q-extensions of Dougall’s bilateral ${_2H_2}$-series, Ramanujan J. 25 (1), 121–139, 2011.
- [12] W. Chu, Evaluation of nonterminating hypergeometric $_3F_2(\frac34)$-series, J. Math. Anal. Appl. 450 (1), 490–503, 2017.
- [13] W. Chu and X. Wang, The modified Abel lemma on summation by parts and terminating hypergeometric series identities, Integral Transforms Spec. Funct. 20 (2), 93–118, 2009.
- [14] W. Chu, X. Wang, and D.Y. Zheng, Application of the residue theorem to bilateral hypergeometric series, Matematiche 62 (2), 127–146, 2007.
- [15] A.C. Dixon, Summation of a certain series, Proc. London Math. Soc. 35 (1), 284–291, 1903.
- [16] M.A. Dougall, On Vandermonde’s theorem and some more general expansion, Proc. Edin. Math. Soc. 25, 114–132, 2007.
- [17] R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley Publ. Company, Reading, Massachusetts, 1989.
- [18] M. Jackson, A note on the sum of a particular well–poised $_6H_6$ with argument −1, J. London Math. Soc. 27, 124–126, 1952.
- [19] K. Knopp, Theory and Applications of Infinite Series, Hafner Publishing Company, New York, 1971.
- [20] E.D. Rainville, Special Functions, New York, The Macmillan Company, 1960.
- [21] H.M. Srivastava, Y. Vyas, and K. Fatawat, Extensions of the classical theorems for very well–poised hypergeometric functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113 (367), 2019, https://doi.org/10.1007/s13398-017-0485-5.
- [22] K.R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth, INC. Belmont, California, 1981.
- [23] C. Wang and X. Chen, A short proof for Gosper’s $_7F_6$-series conjecture, J. Math. Anal. Appl. 422 (2), 819–824, 2015.
- [24] C.Wang, J. Dai, and I. Mezo, A nonterminating $_7F_6$-series evaluation, Integral Transforms Spec. Funct. 29 (9), 719–724, 2018.
- [25] F.J.W. Whipple, On well–poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. 24 (2), 247–263, 1926.
Toplam 25 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik |
| Yazarlar | |
| Yayımlanma Tarihi | 6 Ekim 2020 |
| Yayımlandığı Sayı | Yıl 2020 Cilt: 49 Sayı: 5 |