Generalizations of Ramanujan integral associated with infinite Fourier cosine transforms in terms of hypergeometric functions and its applications

2018/05/06 by M. I. Qureshi, Qureshi, M. I., Showkat Ahmad Dar +1
Mathematics · #33C20 #33C60 #33E20 #42A38 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #F.2.2 #FOS: Mathematics #I.2.7 #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1805.02148

openalex publication_date 2018/05/06 · openalex created_date 2018/05/17 · openalex updated_date 2026/07/28

Abstract

In this paper, we obtain analytical solution of an unsolved integral R_C(m,n) of Srinivasa Ramanujan [Mess. Math., XLIV, 75-86, 1915], using hypergeometric approach, Mellin transforms, Infinite Fourier cosine transforms, Infinite series decomposition identity and some algebraic properties of Pochhammer's symbol. Also we have given some generalizations of the Ramanujan's integral R_C(m,n) in the form of integrals I^*_C(υ,b,c,λ,y), J_C (υ,b,c,λ,y), K_C (υ,b,c, λ,y), I_C(υ,b,λ,y) and solved it in terms of ordinary hypergeometric functions ₂ F₃, with suitable convergence conditions. Moreover as applications of Ramanujan's integral R_C(m,n), the new nine infinite summation formulas associated with hypergeometric functions ₀F₁, ₁F₂ and ₂F₃ are obtained.

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