Hypergeometric Forms of Well Known Partial Fraction Expansions of Some Meromorphic Functions

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1 Global Journal of Science Frontier Research Volume Issue 6 Version.0 September 20 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN : & Print ISSN: Hypergeometric Forms of Well Known Partial Fraction Expansions of Some Meromorphic Functions By M.I. Qureshi Izharul H. Khan M.P.Chaudhary Amity Institute of Biotechnology Amity University NOIDA (U.P.) India Abstract - In this paper we obtain hypergeometric forms of some meromorphic functions e z sec2 z cosec 2 z tan z cot z cosec z sec z sech 2 (z) cosech 2 (z) tanh(z) coth(z) cosech(z) sech(z) 8z 3 sinh(2z)+sin(2z) cosh(2z) cos(2z) corresponding partial fraction expansions. z sinh(2z) sin(2z) cosh(2z) cos(2z) and z 2 Keywords : Maclaurin expansion Mittag-Leffler theorem Bernoulli and Euler numbers Ramanujan s transcendental functions. 200 A.M.S. Subject Classifications: 33-Special Functions. sinh(2z) cosh(2z) cos(2z) from Hypergeometric Forms of Well Known Partial Fraction Expansions of Some Meromorphic Functions Strictly as per the compliance and regulations of: 20. M.I. Qureshi Izharul H. Khan M.P.Chaudhary.This is a research/review paper distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License permitting all non-commercial use distribution and reproduction in any medium provided the original work is properly cited.

2 Hypergeometric Forms of Well Known Partial Fraction Expansions of Some Meromorphic Functions M.I. Qureshi α Izharul H. Khan Ω M.P.Chaudhary β Abstract - In this paper we obtain hypergeometric forms of some meromorphic functions e z sec2 z cosec 2 z tan z cot z cosec z sec z sech 2 (z) cosech 2 (z) tanh(z)coth(z) cosech(z) sech(z) sinh(2z) sin(2z) and sinh(2z) from corresponding partial fraction expansions. z cosh(2z) cos(2z) z 2 cosh(2z) cos(2z) Keywords : Maclaurin expansion Mittag-Leffler theorem Bernoulli and Euler numbers Ramanujan s transcendental functions. I. INTRODUCTION In the monumental work of Prudnikov et al.[8chapter 7] and other literature of Special functions hypergeometric forms of following functions sin zcos z sin 2 z cos 2 z sinh(z) cosh(z) sinh 2 z cosh 2 z sin z (sin z) 2 cos z sec zcosec z tan z cot sin z z z 2 sinh z (sinh z) 2 cosh z sech z cosech ztanh z coth sinh z z +z 2 log a (±z) ln(±z) e ±z a ±z ( ± z) ±a sin(a sin z) cos(a sin z) z 2 sin(a sin z) where h 0 2 and the notation ( ) stands for Gamma function. Lemma: If a p and n are suitably adjusted real or complex numbers such that associated Pochhammer s symbols are well-defined then we have (a + pn) a( ) a+p ( p n a (2) p) 8z 3 sinh(2z)+sin(2z) cosh(2z) cos(2z) cos(a sin z) z 2 associated composite functions and transcendental functions are available. In the Maclaurin s expansions oftan zcot zcosec z tanh(z) coth( z) cosech(z) and sec zsech(z) the coefficients of z n are associated with Bernoulli numbers and Euler numbers [5] respectively.from Maclaurin s expansions we are unable to obtain their corresponding hypergeometric forms. Now we shall find the hypergeometric forms of tan z cosec z cot z sec z and other associated composite functions by means of corresponding partial fraction expansions obtained by Mittag-Leffler theorem or Fourier series method [5 pp The Pochhammer s symbol or Shifted factorial (h r) is defined by Γ(h + r) if r 0 (h) r Γ(h) h(h + ) (h + r ) if r 2 3. () Γ Mittag - Leffler s expansion theorem[756] (i) Suppose that the only singularities of f( ) except at infinity in the finite plane are the simple poles at the points z a z a 2 z a 3 arranged in order of increasing absolute value that is: a 3 a 2 a > 0 n Global Journal of Science Frontier Research Volume XI Issue VI vvvvvvvvvvvvvvv ersion I September Author α : Department of Applied Sciences and Humanities Jamia Millia Islamia New Delhi-0025 India. Author Ω : Department of Allied Sciences and Computer Applications Amity Institute of Biotechnology Amity University NOIDA (U.P.) India. Author β : International Scientific Research and Welfare Organization New Delhi-008 India. mpchaudhary 2000@yahoo.com. 20 Global Journals Inc. (US)

3 Global Journal of Science Frontier Resear ch Volume XI Issue e VI Version I September Hypergeometric Forms of Well Known Partial Fraction Expansions of Some Meromorphic Functions (ii) Let the residues of f(z) at a a 2 a 3 be b b 2 b 3 respectively. (iii) Let C N be circles of radius R N which do not pass through any poles and on which f(z) < M where M is independent of N and R N as N. When these conditions are satisfied then Mittag-Leffler s expansion theorem states that f(z) f(0) + b n + (3) z a n a n for all values of z except the poles. n In the literature of calculus of residues [2 to 7 2 to 6] following partial fractionexpansions are available. [2 pp p.20(q.no.3) 6 p.3] e z z 2 + 2z z 0 ±2i ±i () n z 2 n [5 p.87] sec 2 z ( 2z) 2 + ( + 2z) 2 + (3 2z) 2 + (3 + 2z) [7 p.35 6 p.3] [(2n + ) + 2z] 2 z ± 2 ±3 2 ±5 2 (5) cosec 2 z z 2 + (z ) 2 + (z + ) 2 + (z 2) 2 + (z + 2) (z n) 2 z 0 ± ±2 ±3 (6) [2 p.296 p.57(q.no.36) 6 p.3] tan z 8z 2 z z z + 2 2z (n + 2 )2 2 z 2 + z ± 2 ±3 2 ±5 2 z (n + ) + (n + ) 2 2 (7) [3 p.22(q.no.8) 5 p p.30(q.no.)] cot z z + 2z z z z z + 2z n z 2 n 2 2 z 0 ± ±2 ±3 (8) 20 Global Journals Inc. (US)

4 [3 p.22(q.no.9) p.7 7 pp.32-33] cosec z z 2z z 2 2 z z z + 2z n ( ) n n 2 2 z 2 z 0 ± ±2 ±3 (9) [ p.56(q.no.3) 5 p p.37(q.no.8)] sec z 2 z z z 2 2 ( ) n (n + 2 ) (n + 2 )2 2 z 2 z ± 2 ±3 2 ±5 2 (0) [2 p p.87] tanh(z) 8z 2 + z z z + 2 2z (n + 2 )2 2 + z 2 z ± i 2 ±3i 2 ±5i 2 () [2 p p.3] coth(z) z +2z z z z z + 2z n z 2 + n 2 2 z 0 ±i ±2i ±3i (2) [2 p p.35] cosech(z) z 2z z z z z + 2z n ( ) n n z 2 z 0 ±i ±2i ±3i (3) Global Journal of Science Frontier Research Volume XI Issue VI vvvvvvvvvvvvvvv ersion I September [ p.75 5 p.87] sech(z) 2 + z z z 2 2 ( ) n (n + 2 ) (n + 2 )2 2 + z 2 z ± i 2 ±3i 2 ±5i 2 () 20 Global Journals Inc. (US)

5 Global Journal of Science Frontier Resear ch Volume XI Issue e VI Version I September Hypergeometric Forms of Well Known Partial Fraction Expansions of Some Meromorphic Functions Ramanujan s partial fraction expansions [ Part -IV pp ] Ramanujan s systematic work on ordinary hypergeometric series is contained primarily in Chapters XII XIII XV of first notebook [9] and Chapters III X and XI of second notebook [0]. Ramanujan evidently had an affinity for partial fraction expansions which can be found in several places in his notebooks. The heaviest concentrations lie in Chapters and 8 and in the unorganized pages at the end of the second notebook. See Berndt s books [part-ii] and [part-iii] for accounts of the material in Chapters and 8 respectively. In this paper we obtain the hypergeometric forms of three partial fraction decompositions in the unorganized pages of second notebook. When z m (±i) m 0 ± ±2 ±3 2 then [(Part-IV) pp Entry 3 see also 7 p.37 (Q.No.20 i)] sinh(2z) + sin(2z) 8z 3 cosh(2z) cos(2z) 8z + (5) z + n [(Part-IV) pp Entry ] [(Part-IV) pp Entry 5] II. z 2 z sinh(2z) sin(2z) cosh(2z) cos(2z) n HYPERGEOMETRIC FORMS OF SOME PARTIAL FRACTION EXPANSIONS If we apply the Lemma (2) in real or complex linear factors of quadratic and biquadratic polynomials in n associated with the denominators of partial fraction expansions () to (7) we get the following hypergeometric forms: sec 2 z e z z 2 + sec 2 z n sinh(2z) cosh(2z) cos(2z) 8z + n 3 z + n + 2z (2z + ) 2 2 H 2 (2z ) 2 3 F 2 cosec 2 z 2z (z ) 3 F 2 3+2z 2 2z 2z z 2 +2z +2z z 2 3+2z 2 cosec 2 z z z z 2 2 H 2 z z (z + ) 2 3 F 2 +z +z 2+z n 2 z + n (6) n 2+iz 2 iz 2 2 +iz 2 iz 2 n z 2 + (z + n) 2 (7) z 0 ±2i (8) z ± 2 ±3 2 ±5 2 (9) + (2z + ) 2 3 F 2 2+z +2z +2z z 2 3+2z 2 (20) z 0 ± ±2 ±3 (2) + z z z 2 3 F 2 z z (22) 20 Global Journals Inc. (US)

6 tan z 8z ( 2 z 2 ) 3 F 2 cot z z + 2z (z 2 2 ) 3 F 2 cosec z z + 2z ( 2 z 2 ) 3 F 2 sec z ( 2 z 2 ) F 3 +2z 2z z 2 3 2z 2 +z z 2+z 2 z +z z 2+z 3 +2z z z z ± 2 ±3 2 ±5 2 (23) By replacing by in (9) to (26) and using the identities sec( iz) sech(z) cosec (iz) i cosech(z)tan (iz) i tanh(z)cot(iz) i coth(z) we get the hypergeometric forms of corresponding hyperbolic functions. sech 2 (z) sech 2 (z) (2iz + ) 2 2 H 2 ( 2iz) 2 3 F 2 cosech 2 (z) z 2 2 H 2 2z 2 3 2z 2 +2iz +2iz iz 2 3+2iz 2 2iz 2iz iz 2 iz iz iz 3 2iz 2 iz cosech 2 (z) iz iz z 2 3 F 2 iz iz tanh(z) 8z ( 2 + z 2 ) 3 F 2 coth(z) z + 2z (z ) 3 F 2 z 0 ± ±2 (2) z 0 ± ±2 (25) z ± 2 ±3 2 (26) + ( + 2iz) 2 3 F 2 z ± i 2 ±3i 2 (27) +2iz +2iz iz 2 3+2iz 2 (28) z 0 ±i ±2i ±3i (29) +2iz 2iz iz 2 3 2iz 2 +iz iz 2+iz 2 iz ( + iz) 2 3 F 2 +iz +iz 2+iz 2+iz (30) z ± i 2 ±3i 2 (3) z 0 ±i ±2i (32) Global Journal of Science Frontier Research Volume XI Issue VI vvvvvvvvvvvvvvv ersion I September Global Journals Inc. (US)

7 cosech(z) z 2z ( 2 + z 2 ) 3 F 2 +iz iz 2+iz 2 iz z 0 ±i ±2i (33) Global Journal of Science Frontier Resear ch Volume XI Issue e VI Version I September sech(z) ( 2 + z 2 ) F iz iz 2 2 When z m ( ± i) m 0 ± ±2 ±3 2 then 8z 3 sinh(2z) + sin(2z) cosh(2z) cos(2z) 8z + Above hypergeometric forms are not available in the literature. It is to be noted that the hypergeometric series 3 F 2 F 3 5 F and 6 F 5 are convergent. III. 2iz 2 3 2iz 2 PROOFS To derive (8) consider the following partial fraction expansion z ± i 2 ±3i 2 (3) (z + ) z + + iz z + iz z + + iz z + iz 5 F (35) z iz z + 2 iz z iz z + 2 iz sinh(2z) sin(2z) z cosh(2z) cos(2z) (z + ) 2 2 z + + iz z + iz z + + iz z + iz 6 F 5 (36) z iz z + 2 iz z iz z + 2 iz sinh(2z) z 2 cosh(2z) cos(2z) 8z z + + iz z + iz z + + iz z + iz + (z + ) 5 F + z iz z + 2 iz z iz z + 2 iz z + + iz z + iz + (z 3 + z 2 + 2z) 3 F 2 (37) z iz z + 2 iz e z z 2 + 2z z 0 ±2i ±i ±6i z 2 + n 2 2 z 2 + 2z z 2 +2z n [z + 2i(n + )] [z 2i(n + )] [(z + 2i) + (2i)n] [(z 2i) + ( 2i)n] 20 Global Journals Inc. (US)

8 Now using the beautiful Lemma (2) we get e z z 2 + 2z To derive (2) and (22) consider the following expansion which is the hypergeometric form (2). Now replacing z by iz and using suitable circular-hyperbolic identity we get (29). Now again consider Now using the Lemma (2) we get (z + 2i) ( z+i 2i z 2 + 2z (z ) 3 F 2 e z z 2 + cosec 2 z + + 2z (z ) 3 F 2 ( z+2i) ( z 2i) 2i n 2i n ) n z i (z 2i) ( ) 2i n 2i+z 2i z 2i 2i i+z 2i i z 2i 2+iz 2 iz 2 2 +iz 2 iz 2 (z n) 2 z 0 ± ±2 ±3 [z + ( )n] [z + ( )n] Now using the beautiful Lemma (2) we get + ( z cosec 2 z ) ( z ) n n z ( z ) z ( z ) z z z n 2 2 H 2 z z n cosec 2 z (z n) (z n) 2 z 0 ± ±2 ±3 (z n) 2 [(z + ) + n] 2 + n (z + n) 2 + [z + ( )n] 2 (z n) 2 Global Journal of Science Frontier Research Volume XI Issue VI vvvvvvvvvvvvvvv ersion I September cosec 2 z () n ( z+ )2 n (z + ) 2 ( z+2 )2 n! + n (z + ) 2 3 F 2 +z +z 2+z 2+z () n ( z )2 n z 2 ( z + )2 n n! z 2 3 F 2 z z z z 20 Global Journals Inc. (US)

9 which is the hypergeometric form (22). Now replacingz by iz we get (30). Similarly we can obtain the hypergeometric forms (9) (20) (23) to (28) (3) to (37) of remaining partial fraction expansions. Global Journal of Science Frontier Resear ch Volume XI Issue e VI Version I September IV. ACKNOWLEDGEMENT The authors are thankful to Prof. R. Y. Denis and Prof. M. A. Pathan for their valuable suggestions during the preparation of this research paper. The corresponding author (MPC) is also thankful to the Library Staff and Scientists of CRM Marseille France EUROPE for their cooperation during his academic visit in winter 200. REFERENCES RÉFÉRENCES REFERENCIAS. Berndt B. C. Ramanujan s notebooks. Parts I-V Springer-Verlag New York Bromwich T. J. I A. An introduction to the theory of infinite series. Second Edition. Macmillan and Co Conway J. B. Functions of one complex variable. Narosa Publishing House New Delhi Bombay Calcutta Second Edition Copson E. T. An introduction to the theory of function of a complex variable. Oxford Univ. Press London and New York Geoffrey Cum Berlege Courant R. and John F. Introduction to calculus and analysis Volume I. Springer-Verlag New York Inc First Indian Reprint Kyrala A. Applied functions of a complex variable. John Wiley & Sons Inc. New York London Toronto Phillips E. G. Functions of a complex variable with applications. Longman Group Limited London Prudnikov A. P. Brychkov Yu. A. and Marichev O. I. Integrals and series Vol. 3: More special functions. Nauka Moscow 986. Translated from the Russian by G. G. Gould Gordon and Breach Science Publishers New York Philadelphia London Paris Montreux Tokyo Melbourne Ramanujan S. Notebooks of Srinivasa Ramanujan. Vol. I Tata Institute of Fundamental Research Bombay 957 Reprinted by Narosa New Delhi Ramanujan S. Notebooks of Srinivasa Ramanujan. Vol. II Tata Institute of Fundamental Research Bombay 957 Reprinted by Narosa New Delhi98.. Ramanujan S. The Lost notebook and other unpublished papers. Narosa PublishingHouse New Delhi Rudin W. Real and complex analysis. McGraw-Hill Inc. New York Slater L. J. Generalized hypergeometric functions. Cambridge Univ. Press Cambridge London and New York Spiegel M. R. Theory and problems of complex variables with an introduction to conformal mapping and its application. Schaum s outline series McGraw- Hill Book Company SI (Metric) Edition Singapore Spiegel M. R. Mathematical handbook of formulas and tables. Schaum s outline series McGraw-Hill Publishing Company New York Titchmarsh E. C. The theory of functions. Oxford University Press Ely Edition 939. House London Second 20 Global Journals Inc. (US)