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Kyungpook Mathematical Journal 2020; 60(4): 781-795

Published online December 31, 2020

Copyright © Kyungpook Mathematical Journal.

Generalizations of Ramanujan’s Integral Associated with Infinite Fourier Cosine Transforms in Terms of Hypergeometric Functions and its Applications

Mohammad Idris Qureshi, Showkat Ahmad Dar*

Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia(A Central University), New Delhi, 110025, India
e-mail : miqureshi_delhi@yahoo.co.in
Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia(A Central University), New Delhi, 110025, India
Post Graduate Department of Mathematics, Govt. Degree College Boys Baramulla, University of Kashmir, Kashmir, 193502, India
e-mail : showkatjmi134@gmail.com

Received: September 22, 2018; Revised: July 24, 2020; Accepted: July 28, 2020

Abstract

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In this paper, we obtain an analytical solution for an unsolved definite integral R C ( m , n ) from a 1915 paper of Srinivasa Ramanujan. We obtain our solution using the hypergeometric approach and an infinite series decomposition identity. Also, we give some generalizations of Ramanujan's integral R C ( m , n ) defined in terms of the ordinary hypergeometric function 2F3 with suitable convergence conditions. Moreover as applications of our result we obtain nine new infinite summation formulas associated with the hypergeometric functions 0F1 , 1F2 and 2F3.

Keywords: generalized hypergeometric function, infinite Fourier cosine transforms, Ramanujan’s integrals, Fox-Wright psi hypergeometric function, Mellin transforms, series decomposition identity, bounded sequence

1. Introduction and Preliminaries

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In the literature of infinite Fourier cosine transforms (see, for example, [2, 4]) one can find analytical solutions of ∫0∞xυ−1cos(xy)exp(bx)±1dx given in terms of Riemann's zeta function, the Psi (Digamma) function, hyperbolic functions and Beta functions.

The analytical solution of the following integral of Ramanujan [7, p. 85, eq.(49)]:

RC(m,n)=∫0∞ xmcos(πnx) exp(2πx)−1dx,

is not known for all positive rational values of n, and non-negative integral values of m; though for three particular pairs or values of m and n the following solutions are given in [7, p.86, eq.(50)]:

RC(1,1/2) =∫0∞ xcos( πx2) exp(2πx)−1dx=13−4π8π2 , RC(1,2) =∫0∞ xcos(2πx) exp(2πx)−1dx=16412−3π+5π2 , RC(2,2) =∫0∞ x2cos(2πx) exp(2πx)−1dx=12561−5π+5π2 .

The following theorem is proved by Ramanujan [7, p.76-77, eqs.(10 and 10')]: If

RC(0,n)=Φ(n)=∫0∞ cos(πnx) exp(2πx)−1dx,

and

  ϒ(n)=12πn+∫0∞ sin(πnx) exp(2πx)−1 dx,

then

RC(0,n)=Φ(n)=1n2n Υ1n−Υ(n),

and

ϒ(n)=1n2n Φ1n+Φ(n),    

where n is positive rational number.

For particular values of n, Ramanujan [7, p.85, eq. (48)] also showed the following:

RC(0,1) =Φ(1)=∫0∞ cos(πx) exp(2πx)−1dx=2−2 8, RC(0,2) =Φ(2)=∫0∞ cos(2πx) exp(2πx)−1dx=116, RC(0,4) =Φ(4)=∫0∞ cos(4πx) exp(2πx)−1dx=3−2 32, RC(0,6) =Φ(6)=∫0∞ cos(6πx) exp(2πx)−1dx=13−43 144, RC(0,1/2) =Φ12=∫0∞ cos πx 2 exp(2πx)−1 dx=14π, RC(0,2/5) =Φ25=∫0∞ cos 2πx 5 exp(2πx)−1 dx=8−3516.

A natural generalization of Gauss hypergeometric series 2F1 is the general hypergeometric series pFq [9, p.42, eq.(1)] and see also [1] with p numerator parameters α1,...,αp and q denominator parameters β1,...,βq. For p,q∈ℕ0:=ℕ∪{0}={0,1,2,...}, it is defined as

pFq  α1 ,...,αp  ;  β1 ,...,βq  ; z=∑ n=0∞ (α1 )n ...(αp )n (β1 )n ...(βq )n znn! ,

where αi∈ℂ for i=1,...,p and βj∈ℂ∖ℤ0−. Here we use ℤ0−:={0,−1,−2,...}. Also for λ,υ∈ℂ, Pochhammer's symbol (λ)υ,(or the shifted factorial, since (1)n=n!) is defined, in general, by

(λ)υ:=Γ(λ+υ)Γ(λ)=1,              (υ=0 ; λ∈ℂ\{0})λ(λ+1)...(λ+n−1), (υ=n∈ℕ ; λ∈ℂ).

The hypergeometric series pFq in eq.(1.5) is convergent for |z|<∞ if p≤q, and for |z|<1 if p=q+1.

Furthermore, if we set

ω= ∑ j=1qβj−∑ i=1pαi,

it is known that the series pFq, with p=q+1, is

  • (i) absolutely convergent for |z|=1 if Re(ω)>0,

  • (ii) conditionally convergent for |z|=1,z≠1, if −1<Re(ω)≤0.

Also the binomial function is given by

(1−z)−α=1F0α ;   ¯;z=∑ n=0∞ (α)n n!zn ,

where |z|<1, α∈ℂ∖ℤ0−.

The Fox-Wright function pΨq from [11, 12] is given by

pΨq(α1 ,A1 ),...,(αp ,Ap );(β1 ,B1 ),...,(βq ,Bq ); z=∑ k=0∞Γ(α1 +kA1 )...Γ(αp +kAp )Γ(β1 +kB1 )...Γ(βq +kBq )zkk!, =Γ(α1)...Γ(αp)Γ(β1)...Γ(βq)∑ k=0∞(α1)kA1...(αp)kAp(β1)kB1...(βq)kBqzkk!,= Γ( α 1 )...Γ( α p ) Γ( β 1 )...Γ( β q ) pΨq*(α1,A1),...,(αp,Ap); (β1,B1),...,(βq,Bq);  z, =12πρ∫L Γ(ζ)∏i=1pΓ(αi −Ai ζ) ∏j=1qΓ(βj −Bj ζ) (−z)−ζdζ ,

where we have ρ=−1,  , and z,αi,βj∈ℂ everywhere; and Ai,Bj∈ℝ∖0 except in the case of (1.21) where we have Ai,Bj∈ℝ+=(0,+∞) In eq. (1.19), the parameters αi,βj and coefficients Ai,Bj are adjusted in such a way that the product of the Gamma functions in numerator and denominator should be well defined.

Suppose:

Δ* = ∑ j=1qBj−∑ i=1pAi, δ* =∏ i=1p|Ai|−Ai∏ j=1q|Bj|Bj, μ* =∑ j=1qβj−∑ i=1pαi+p−q2,

and

σ*=(1+A1+...+Ap)−(B1+...+Bq)=1−Δ*.

Then we have the following convergence conditions of (1.19) and (1.21):

Case(1): When the contour (L) is a left loop beginning and ending at -∞, then pΨq[⋅], given by (1.19) or (1.21), converges under any of the following conditions.

  • i) Δ*>−1, and 0<|z|<∞.

  • ii) Δ*=−1 and 0<|z|<δ*.

  • iii) Δ*=−1, |z|=δ*, and Re(μ*)>12.

Case(2): When the contour (L) is a right loop beginning and ending at +∞, then pΨq[⋅], given by (1.19)or (1.21), converges under the following conditions.

  • iv) Δ*<−1, and 0<|z|<∞.

  • v) Δ*=−1, and |z|>δ*.

  • vi) Δ*=−1, |z|=δ*, and Re(μ*)>12.

Case(3): When contour (L) is starts at γ−i∞ and ends at γ+i∞ where γ∈ℝ, then pΨq[⋅] is also convergent under the following conditions.

  • vii) σ*>0, |arg(−z)|<π2σ*, and 0<|z|<∞.

  • viii) σ*=0, arg(−z)=0, 0<|z|<∞ and −γΔ*+Re(μ*)>12+γ.

  • ix) γ=0, σ*=0,arg(−z)=0, 0<|z|<∞, and Re(μ*)>12.

The infinite Fourier cosine transform of g(x) over the interval (0,∞) is defined by

FC{g(x);y}=∫0∞g(x)cos(xy)dx=GC(y),     (y>0).

It follows that we have g(x)=FC−1[GC(y);x]=2π∫0∞GC(y)cos(xy)dy.

Note that some authors add an extra factor of 2π in their definition of FC{g(x);y}.}

If b>0 and 0<Re(s)<1, then the Mellin-transform of cos(bx) or infinite Fourier Cosine transform of xs−1 [3, p.42, eqs.(5.2)] is given by

∫0∞ x s−1cos(bx)dx=Γ(s)cos(πs2)bs .

If Re(μ)>−1, 0<ξ<1, a > 0 and y > 0, then we can prove the following integral using Maclaurin's expansion of exp(−axξ) and integrating termwise with the help of the result (1.27):

∫0∞ xμexp(−axξ)cos(xy)dx=−y−μ−1∑ l=0∞−ayξ l 1l!Γ(μ+1+ξl) sinπ2(μ+ξl).

An infinite series decomposition identity [8, p.193,eq.(8)]is given by

∑ l=0∞Ω(l)=∑ j=0 N−1∑l=0∞Ω(Nl+j),

where N is an arbitrary positive integer. Put N=4 in the above eq. (1.29), we get

∑ l=0∞Ω(l)=∑ j=03∑l=0∞Ω(4l+j),   =∑ l=0∞Ω(4l)+∑ l=0∞Ω(4l+1)+∑ l=0∞Ω(4l+2)+∑ l=0∞Ω(4l+3),

provided that all involved infinite series are absolutely convergent.

For every positive integer m [9, p.22, eq.(26)], we have

(λ)mn=mmn∏ j=1 mλ+j−1 mn        ;m∈ℕ, n∈ℕ0.

From the above result (1.32) with λ=mz, it can be proved that

Γ(mz)=(2π)(1−m)2mmz−12∏ j=1 mΓz+j−1m,   mz∈ℂ\ℤ0−

Equation (1.33) is known as the Gauss-Legendre multiplication theorem for the Gamma function. Elementary trigonometric functions [9,p.44, eq.(9) and eq.(10)] are given by

cos z=0F1 −;    12; −z24,  sin z=z 0F1 −;    32; −z24.

The Lommel function [9, p.44, eq.(13)] is given by

sμ,υ(z)=zμ+1(μ−υ+1)(μ+υ+1)1F2                 1; μ−υ+32,μ+υ+32; −z24,

where μ±υ∈ℂ∖{−1,−3,−5,−7,...}.

As we have mentioned, no general analytic solution is known for RC(m,n). Motivated by the work done in [10, 5] our aim in this paper is to give an analytical solution of Ramanujan's integral in terms of ordinary hypergeometric functions.

Here in this paper, we have generalized Ramanujan's integral RC(m,n) in the following forms, where {Θ(k)}k=0∞ is a bounded sequence, and obtain analytical solution for them:

  • (i) IC*(υ,b,c,λ,y)=∑k=0∞[Θ(k)k!∫0∞ xυ−1e−(λb+ck)xcos(xy)dx],

  • (ii) JC(υ,b,c,λ,y)=∫0∞xυ−1e−bλxrΨs (α1,A1),...,(αr,Ar);(β1,B1),...,(βs,Bs); e−cxcos(xy)dx,

  • (iii) KC(υ,b,c,λ,y)=∫0∞xυ−1e−bλxrFs α1,...,αr;β1,...,βs; e−cxcos(xy)dx,

  • (iv) IC(υ,b,λ,y)=∫0∞xυ−1{exp(bx)−1}−λcos(xy)dx

Moreover, we show, in Sections 3-6, how the main general theorem given below can be applied to obtain new interesting results by suitably adjusting the parameters and variables.

2. Main General Theorem on In nite Fourier Cosine Transform

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Suppose {Θ(k)}k=0∞ is a bounded sequence of arbitrary real and complex numbers, and and Re(υ),c,y, are positive and λ and b are both positive or both negative, then

IC*(υ,b,c,λ,y)=∑ k=0∞ Θ(k) k! ∫0 ∞ x υ−1e −(λb+ck)x cos(xy)dx, =y−υ∑ k=0∞[Θ(k)k!∑ l=0∞(−1)l(λb+ck)lΓυ+l2 yl2  l!cosυπ2+lπ4],

Now replacing ℓ by 4ℓ + j, after simplification we get

IC*(υ,b,c,λ,y)=y−υ∑ k=0∞[Θ(k)k!∑ j=03(−1)j(λb+ck)j Γυ+j2yj2 j!cosυπ2+jπ4×2F3Δ2; 2υ+j2 ; Δ*4;1+j; −164y2(λb) ( λb+cc )k ( λbc )k 4], =y−υ∑ k=0∞[Θ(k)k!∑ j=03(−1)jΓυ+j2j!cosυπ2+jπ4 λb y j×× (λb+cc)k (λbc)k j2F3Δ2; 2υ+j2 ; Δ* 4;1+j; −164y2 (λb) ( λb+c c ) k ( λb c ) k 4], =Γ(υ)cosυπ2yυ∑ k=0∞[ Θ(k) k! 2F3υ2,υ+12 ;14, 12, 34;−164y2(λb)(λb+cc )k (λbc )k 4]−(λb)Γ(υ+12)cosυπ2+π4yυ+12∑ k=0∞[Θ(k)k! (λb+cc)k(λbc)k××2F32υ+14,2υ+34;12, 34, 54     ;−164y2(λb)(λb+cc )k (λbc )k 4]−(λb)2Γ(υ+1)sinυπ22yυ+1∑ k=0∞[Θ(k)k! ( λb+c c ) k ( λb c ) k 2××2F3υ+12,υ+22;34, 54, 32  ;−164y2(λb)(λb+cc )k (λbc )k 4]+(λb)3Γ(υ+32)sinυπ2+π46yυ+32∑ k=0∞[Θ(k)k! ( λb+c c ) k ( λb c ) k 3××2F32υ+34,2υ+54;54, 32, 74     ;−164y2(λb)(λb+cc )k (λbc )k 4]

Our result (2.3) or (2.4) or (2.5) is convergent in view of the convergence condition of pFq(⋅), when p≤q, and ∀ |z|<∞.

3. Infinite Fourier Cosine Transforms of
x υ − 1 e − b λ x r Ψ s [ ⋅ ] and
x υ − 1 e − b λ x r F s ( ⋅ )

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If we put Θ(k)=Γ(α1+kA1)...Γ(αr+kAr)Γ(β1+kB1)...Γ(βs+kBs), for k=0,1,2,3,...,

in the equations (2.1) and (2.3), then after simplification we get the following:

JC(υ,b,c,λ,y)=∫0∞ x υ−1e−bλx rΨs (α1,A1),...,(αr,Ar);(β1,B1),...,(βs,Bs); e−c x cos(xy)dx, =y−υ∑ k=0∞[Γ(α1+kA1)...Γ(αr+kAr)Γ(β1+kB1)...Γ(βs+kBs)k!∑ j=03(−1)j(λb+ck)j Γυ+j2yj2 j!××cos υπ2 + jπ4 2F3Δ2; 2υ+j2 ; Δ*4;1+j; −164y 2(λb) ( λb+cc )k ( λbc )k 4],

where Re(υ),c,y, are positive, λ and b are both positive or both negative, αi,βj∈ℂ and Ai,Bj∈ℝ∖{0} for i=1,2,...,r and j=1,2,...,s, and rΨs[⋅] is the Fox-Wright psi function of one variable subject to suitable convergence conditions derived from the convergence conditions for (1.19),(1.20) and (1.21) given in Case(1) or Case(2) or Case(3).

When N is a positive integer then Δ(N;λ) denotes the array of N parameters given by λN,λ+1N,...,λ+N−1N. When N and j are independent variables then the notation Δ(N;j+1) denotes the set of N parameters given by j+1N,j+2N,...,j+NN. When j is dependent variable that is j=0,1,2,3,...,N−1, then the asterisk in Δ*(N;j+1) represents the fact that the (denominator) parameters NN is always omitted (due to the need of factorial in denominator in the power series form of hypergeometric function) so that the set Δ*(N;j+1) obviously contains only (N-1) parameters [9, Chap.3, p.214].

Remark 3.1. When A1=...=Ar=B1=...=Bs=1 in (3.1), (3.2) then we get

KC(υ,b,c,λ,y)=∫0∞ x υ−1e−bλx rFs α1,...,αr;β1,...,βs; e−c x cos(xy)dx, =y−υ∑ k=0∞[(α1)k...(αr)k(β1)k...(βs)k k!∑ j=03(−1)j(λb+ck)j Γυ+j2yj2 j!cosυπ2+jπ4×2F3Δ2; 2υ+j2 ; Δ*4;1+j; −164y2(λb) ( λb+cc )k ( λbc )k4],

where Re(υ),c,y, are positive, λ and b are both positive or both negative, r≤s+1, and αi,βj∈ℂ for i=1,2,...,r and j=1,2,...,s.

4. Fourier Cosine Transform of
x υ − 1 { exp ( b x ) − 1 } − λ

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For the generalization IC(υ,b,λ,y) of Ramanujan's integral RC(m,n) in terms of ordinary hypergeometric functions 2F3, the following holds:

IC(υ,b,λ,y)=∫0∞ x υ−1cos(xy) {exp(bx)−1}λdx,      =y−υ∑ k=0∞[ (λ)kk!∑ l=0∞(−1)l(λb+bk)lΓυ+l2 yl2  l!cosυπ2+lπ4], =y−υ∑ k=0∞[(λ)kk!∑ j=03(−1)j(λb+bk)j Γυ+j2yj2 j!cosυπ2+jπ4×2F3Δ2; 2υ+j2 ; Δ*4;1+j; −164y2(λb) (λ+1)k (λ)k 4], =Γ(υ)cosυπ2yυ∑ k=0∞[ (λ)k k! 2F3υ2,υ+12 ;14, 12, 34;−164y2(λb)(λ+1) k (λ) k4]−(λb)Γ(υ+12)cosυπ2+π4yυ+12∑ k=0∞[ (λ+1)k k! 2F32υ+14,2υ+34;12, 34, 54   ;−164y2(λb)(λ+1) k (λ) k4]−(λb)2Γ(υ+1)sinυπ22yυ+1∑ k=0∞[ (λ+1)k 2 (λ)k k! 2F3υ+12,υ+22;34, 54, 32 ;−164y2(λb)(λ+1) k (λ) k4]+(λb)3Γ(υ+32)sinυπ2+π46yυ+32∑ k=0∞[ (λ+1)k 3 k! (λ)k 2 2F32υ+34,2υ+54;54, 32, 74   ;−164y2(λb)(λ+1) k (λ) k4],

where Re(υ),y,λ,b>0.

Proof. In eq.(2.1), put Θ(k)=(λ)k and c=b, we obtain

IC(υ,b,λ,y)=∫0∞ xυ−1e−(λb)x ∑ k=0 ∞(λ)kk! e−(bk)x cos(xy)dx.

Using the binomial expansion (1.18) in (4.5), after simplification we get (4.1). Equations (4.2), (4.3) and (4.4) are obtained from (2.2), (2.3) and (2.5) by putting Θ(k)=(λ)k and c=b.

5. Ramanujan's Integral
R C ( m , n )

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The analytical solution of the integral RC(m,n) is given by

RC(m,n)=∫0∞ xmcos(πnx) exp(2πx)−1dx,   =−(nπ)−m−1∑ k=0∞[∑ l=0∞1l!−(2π+2πk)nπlΓm+1+l2sinmπ2+lπ4], =−(nπ)−m−1∑k=0∞[∑j=031j!−(2π+2πk)nπjΓm+1+j2sinmπ2+jπ4×2F3Δ2;2m+j+22;Δ*4;1+j;−π24n2(2)k(1)k4], =−m! sinmπ2(nπ)m+1∑k=0∞[2F3m+12,m+22;14, 12, 34   ;−π24n2 (2)k (1)k 4]+32msinmπ2+π4(π)m(n)m+32∑ k=0∞[ (2)k (1)k 2F32m+34,2m+54;12, 34, 54      ;−π24n2 (2)k (1)k 4]−(2)(m+1)!cosmπ2(π)m(n)m+2∑ k=0∞[ (2)k (1)k 22F3m+22,m+32;34, 54, 32   ;−π24n2 (2)k (1)k 4]+52mcosmπ2+π4(π)m−1(n)m+52∑ k=0∞[ (2)k (1)k 32F32m+54,2m+74;54, 32, 74      ;−π24n2 (2)k (1)k 4],

where m is a non-negative integer and n is positive rational number.

Proof. The results (5.1), (5.2), (5.3) and (5.4) are obtained from (4.1), (4.2), (4.3) and (4.4) by putting υ=m+1, b=2π, λ=1 and y=nπ.

6. Applications of Ramanujan's Integrals

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In this section we establish the following nine new infinite summation formulas associated with hypergeometric series 0F1, 1F2 and 2F3:

∑k=0∞[2F3 1,32    ;14,12,34;−π2 (2)k (1)k 4]−3π2∑ k=0∞[(2)k(1)k 1F2 74    ;12,34;−π2 (2)k (1)k 4]+5π2∑ k=0∞[ (2) k (1) k 31F2 94    ;54,32;−π2 (2)k (1)k 4]=1324π−13, ∑k=0∞[2F31,32    ;14,12,34;−π216 (2)k (1)k 4]−3π4∑ k=0∞[ (2)k (1)k 1F2 74    ;12,34;−π216 (2)k (1)k 4]+5π28∑ k=0∞[ (2)k (1)k 31F2 94    ;54,32; −π216 (2)k (1)k 4]=π2163π−12−5π2, ∑k=0∞[(2)k(1)k2F3  74 ,94     ; 12 ,34 ,54 ;  −π2 16 (2) k (1) k 4]−165∑k=0∞[(2)k(1)k22F3  2,52     ; 34 ,54 ,32 ; −π2 16 (2) k (1) k 4]+7π6∑k=0∞[(2)k(1)k32F3  94 , 114   ; 54 ,32 ,74 ; −π2 16 (2) k (1) k 4]=π2605π−5 π2 −1, ∑k=0∞[ (2)k (1)k 0F1     ¯; 12;− π24 (2)k (1)k4]−22∑k=0∞[ (2) k (1) k 21F2  1; 34,54; −π24 (2)k (1)k4]+π∑k=0∞[ (2) k (1) k 30F1      ¯; 32 ; −π24 (2)k (1)k4]=2−14, ∑k=0∞[ (2)k (1)k 0F1     ¯; 12; −π2 16 (2)k (1)k4]−2∑k=0∞[ (2) k (1) k 21F2  1; 34,54; −π2 16 (2)k (1)k4]+π2∑k=0∞[ (2) k (1) k 30F1     ¯; 32 ; −π2 16 (2)k (1)k4]=14, ∑k=0∞[ (2)k (1)k 0F1     ¯; 12; −π2 64 (2)k (1)k4]−2∑k=0∞[ (2) k (1) k 21F2  1; 34,54; −π2 64 (2)k (1)k4]+π4∑k=0∞[ (2) k (1) k 30F1      ¯; 32 ; −π2 64 (2)k (1)k4]=32−24, ∑k=0∞[ (2)k (1)k 0F1     ¯; 12; −π2 144 (2)k (1)k4]−233∑k=0∞[ (2) k (1) k 21F2  1; 34,54; −π2 144 (2)k (1)k4]+π6∑k=0∞[ (2) k (1) k 30F1      ¯; 32 ; −π2 144 (2)k (1)k4]=133−1212, ∑k=0∞[ (2)k (1)k 0F1     ¯; 12;−π2 (2)k (1)k4]−4∑k=0∞[ (2) k (1) k 21F2  1; 34,54;−π2 (2)k (1)k4]+2π∑k=0∞[ (2) k (1) k 30F1      ¯; 32 ;−π2 (2)k (1)k4]=18π, ∑k=0∞[ (2)k (1)k 0F1     ¯; 12; −25π2 16 (2)k (1)k4]−25∑k=0∞[ (2) k (1) k 21F2  1; 34,54; −25π2 16 (2)k (1)k4]+5π2∑k=0∞[ (2) k (1) k 30F1      ¯; 32 ; −25π2 16 (2)k (1)k4]=85−15100.

The results (6.1) to (6.3) are obtained by putting m=1,n=12 ; m=1, n=2 and m=2, n=2 in the equations (5.1) and (5.4) and finally comparing with equations (1.2), (1.3) and (1.4). When m=0 with n=1, 2, 4, 6, 12, 25 in the equations (5.1) and (5.4) and comparing with equations (1.9), (1.10), (1.11), (1.12), (1.13) and (1.14), we get the remaining results (6.4) to (6.9) respectively. In view of the hypergeometric functions (1.34), (1.35) and (1.36), we can express the above results (6.4) to (6.9) in terms of cosine, sine and Lommel functions. Our results (1.27) to (6.9) are convergent in view of the convergence condition of pFq(⋅) series, when p≤q, and for all |z|<∞.

7. Conclusion

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Here, we have described some infinite Fourier cosine transforms of Ramanujan. Various Ramanujan integrals, which may be different from those of presented here, can also be evaluated in a similar way. The results established above seem significant. We conclude our observation by remarking that various new results and applications can be obtained from our general theorem by appropriate choice of parameters υ,λ,b,c,y and bounded sequence {Θ(k)}k=0∞ in IC*(υ,b,c,λ,y). This work is in continuation to our earlier work [6] on infinite Fourier sine transforms.

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