Extensions of Watson's theorem and the Ramanujan-Guinand formula
aa r X i v : . [ m a t h . N T ] F e b EXTENSIONS OF WATSON’S THEOREM AND THERAMANUJAN-GUINAND FORMULA
RAHUL KUMAR
Abstract.
Ramanujan provided several results involving the modified Bessel function K z ( x )in his Lost Notebook. One of them is the famous Ramanujan-Guinand formula, equivalentto the functional equation of the non-holomorphic Eiesenstien series on SL ( z ). Recently,this formula was generalized by Dixit, Kesarwani, and Moll. In this article, we first obtaina generalization of a theorem of Watson and, as an application of it, give a new proof ofthe result of Dixit, Kesarwani, and Moll. Watson’s theorem is also generalized in a differentdirection using µ K z ( x, λ ) which is itself a generalization of K z ( x ). Analytic continuation ofall these results are also given. Introduction
On page 253 of his Lost Notebook [11], Ramanujan recorded the following beautiful for-mula. Let z ∈ C . For α, β > αβ = π , √ α ∞ X n =1 σ − z ( n ) n z K z (2 nα ) − p β ∞ X n =1 σ − z ( n ) n z K z (2 nβ )= 14 Γ (cid:16) z (cid:17) ζ ( z ) n β − z − α − z o + 14 Γ (cid:16) − z (cid:17) ζ ( − z ) n β z − α z o , (1.1)where K z ( x ) is the modified Bessel function [19, p. 34] and σ z ( n ) := P d | n d z is the generalizeddivisor function.Theorem 1.1 was rediscovered by Guinand [8] in 1955 which is why sometimes this formulais referred to as Ramanujan-Guinand formula in the literature. Several years after Ramanu-jan, the special case z = 0 of (1.1) was also rediscovered by Koshliakov [13]. For a detailedhistory of the Ramanujan-Guinand formula (1.1) and its implications, see [2].It is well-known that (1.1) is equivalent to the functional equation of the non-holomorphicEisenstein series. For Re( s ) > τ := x + iy, y >
0, the non-holomorphic Eisensteinseries G ( τ, s ) is defined by G ( τ, s ) := 12 X ( m,n ) ∈ Z ( m,n ) =(0 , y s | mτ + n | s .G ( τ, s ) is a non-holomorphic modular function of weight 0 as it is invariant in τ under theusual action of SL ( Z ). Moreover, G ( τ, s ) has analytic continuation in C except a simple poleat s = 1 with residue π . Analytic continuation of this function can be given by its Fourier Mathematics Subject Classification.
Primary 11M06, 33E20; Secondary 33C10.
Keywords and phrases.
Watson’s lemma, Ramanujan-Guinand formula, Poisson summation formula, Gener-alized modified Bessel function. expansion G ( τ, s ) = ζ (2 s ) y s + √ π Γ (cid:0) s − (cid:1) Γ( s ) ζ (2 s − y − s + 4 √ yπ s Γ( s ) ∞ X n =1 σ s − ( n ) n − s K s − (2 πny ) cos(2 πnx ) . (1.2)It is effortless to see that (1.1) follows from (1.2) and by using the modular property of G ( τ, s ), namely, G ( τ, s ) = G (cid:0) − τ , s (cid:1) , τ ∈ H (upper half-plane).Several proofs of (1.1) have been given in the literature, for example, see [2], [3], and [8].The main ingredient in the proofs of [2] and [8] is the following result of Watson [18]: Theorem 1.1.
For Re ( x ) > and Re ( z ) > , we have ∞ X n =1 (cid:16) nx (cid:17) z K z ( nx ) + 12 Γ( z ) − √ πx Γ (cid:18) z + 12 (cid:19) = 2 √ πx z Γ (cid:18) z + 12 (cid:19) ∞ X n =1 x + 4 n π ) z + . (1.3)In [3], Dixit provided another proof of the Ramanujan-Guinand formula (1.1) withoutinvoking Theorem 1.1. For the details of the proof, we refer the reader to [3, Theorem 1.4].The main goal of this paper is to accomplish two different generalizations of Theorem1.1 along with an application of one of the theorem. In the literature, Theorem 1.1 hasbeen generalized in many directions, for example, see [1], [10]. Theorem 1.1 has importantapplication in Number Theory, for example, see [2] and the references therein. The series onthe left-hand side of (1.1) has played an important role in the work of Gupta and the author[9]. Moreover, very recently, Theorem 1.1 is employed nicely in [6, Section 9] to obtain amodular relation involving the generalized Hurwitz zeta function [6, Equation (1.3)] ζ w ( s, a ) := 4 w √ π Γ (cid:0) s +12 (cid:1) ∞ X n =1 Z ∞ Z ∞ ( uv ) s − e − ( u + v ) sin( wv ) sinh( wu )( n u + ( a − v ) s/ dudv, where w ∈ C \{ } , Re( s ) > a ∈ B := { ξ : Re( ξ ) = 1 , Im( ξ ) = 0 } . For the theory of ζ w ( s, a ), we refer the reader to [6].Recently, Dixit, Kesarwani, and Moll [5] provided an elegant generalization of Theorem1.1. To state their result, we first need to define their new generalization of the modifiedBessel function K z ( x ). For z, w ∈ C , and x ∈ C \{ x ∈ R : x ≤ } , the generalized modifiedBessel function is defined by [5, Equation (1.3)] K z,w ( x ) : = 12 πi Z ( c ) Γ (cid:18) s − z (cid:19) Γ (cid:18) s + z (cid:19) F (cid:18) s − z − w (cid:19) F (cid:18) s + z − w (cid:19) s − x − s ds, (1.4)where c := Re( s ) > | Re( z ) | and F ( a ; c ; z ) is the confluent hypergeometric function definedby [15, p. 172, Equation (7.3)] F ( a ; c ; z ) := ∞ X n =0 ( a ) n ( c ) n z n n ! , | z | < ∞ , with ( a ) m := Γ( a + m ) / Γ( a ) for a ∈ C . Here, and throughout the paper, R ( c ) denotes the lineintegral R c + i ∞ c − i ∞ . XTENSIONS OF WATSON’S THEOREM AND THE RAMANUJAN-GUINAND FORMULA 3
It is straightforward to see that for w = 0, K z,w ( x ) reduced to K z ( x ) by using [14, p. 115,Formula 11.1] K z ( x ) = 12 πi Z ( c ) Γ (cid:18) s − z (cid:19) Γ (cid:18) s + z (cid:19) s − x − s ds. For more details on the theory of K z,w ( x ) and its connection in Analytic Number Theoryand Physics, we refer the reader to [5] and [12].The aforementioned generalization of the Ramanujan-Guinand formula is given in thefollowing theorem [5, Theorem 1.4]. Theorem 1.2.
Let z, w ∈ C . Let K z,w ( x ) be defined in (1.4) . For α, β > such that αβ = π , √ α ∞ X n =1 σ − z ( n ) n z e − w K z ,iw (2 nα ) − p β ∞ X n =1 σ − z ( n ) n z e w K z ,w (2 nβ )= 14 Γ (cid:16) z (cid:17) ζ ( z ) (cid:26) β − z F (cid:18) − z w (cid:19) − α − z F (cid:18) − z − w (cid:19)(cid:27) + 14 Γ (cid:16) − z (cid:17) ζ ( − z ) (cid:26) β z F (cid:18) z w (cid:19) − α z F (cid:18) z − w (cid:19)(cid:27) . (1.5)Here we note that authors of [5] did not follow the approach used in [2] or [8] to proveTheorem 1.2. They used the theory of functions reciprocal in a certain kernel. To followthe same approach as in [2] or [8], one needs to first obtain generalization of the Watson’sresult (1.3) which has been missing from the literature. In this work, we fill this gap, that is,we obtain a generalization of the Watson’s result (1.3). Further as an application of it, weprovide a new proof of Theorem 1.2. Our generalization of (1.3) is recorded in the followingtheorem. Theorem 1.3.
Let
Re( x ) > . Define A ( n, z, w, x ) := F (cid:18)
12 + z ; 12 ; w (2 nπ + ix )8 nπ − ix (cid:19) + F (cid:18)
12 + z ; 12 ; w (2 nπ − ix )8 nπ + 4 ix (cid:19) . (1.6) Let w ∈ C and Re( z ) > . Let K z,w ( x ) be defined in (1.4) . Then ∞ X n =1 (cid:16) nx (cid:17) z K z,w ( nx ) + 12 Γ( z ) F (cid:18) z ; 12 ; − w (cid:19) − √ πx Γ (cid:18)
12 + z (cid:19) e − w F (cid:18)
12 + z ; 12 ; − w (cid:19) = √ πx z Γ (cid:18)
12 + z (cid:19) e − w ∞ X n =1 A ( n, z, w, x )( x + 4 n π ) z + . (1.7)We now obtain an extended version of Theorem 1.3 in which the restriction on z is removed. Theorem 1.4.
Let w ∈ C , Re( x ) > , and A ( n, z, w, x ) be defined in (1.6) . Let M > bean integer. Then for Re( z ) > − M , ∞ X n =1 (cid:16) nx (cid:17) z K z,w ( nx ) + 12 Γ( z ) F (cid:18) z ; 12 ; − w (cid:19) − √ πx Γ (cid:18)
12 + z (cid:19) e − w F (cid:18)
12 + z ; 12 ; − w (cid:19) = √ πx z Γ (cid:18)
12 + z (cid:19) e − w " ∞ X n =1 A ( n, z, w, x ) ( x + 4 n π ) z + − M − X m =0 (cid:18) − z − m (cid:19) x m (2 nπ ) z +2 m +1 ) RAHUL KUMAR + M − X m =0 (cid:18) − z − m (cid:19) x m (2 π ) z +2 m +1 ∞ X n =1 A ( n, z, w, x ) n z +2 m +1 . (1.8)As a special case of Theorem 1.4, we get the result of Watson [18, p. 300, Section 3]: Corollary 1.5.
Let
Re( x ) > . Let M > be an integer. Then for Re( z ) > − M , ∞ X n =1 (cid:16) nx (cid:17) z K z ( nx ) + 12 Γ( z ) − √ πx Γ (cid:18)
12 + z (cid:19) = 2 √ πx z Γ (cid:18)
12 + z (cid:19) " ∞ X n =1 ( x + 4 n π ) z + − M − X m =0 (cid:18) − z − m (cid:19) x m (2 nπ ) z +2 m +1 ) + M − X m =0 (cid:18) − z − m (cid:19) x m ζ (2 z + 2 m + 1)(2 π ) z +2 m +1 . (1.9)Next, we present a generalization of Theorem 1.1 in another direction, different fromTheorem 1.3. Very recently, a new two-parameter generalization of K z ( x ) was introducedby Dixit, Kesarwani and the author in [4]. For z ∈ C \ ( Z \{ } ), and x, µ, λ ∈ C such that µ + λ = − , − , − , · · · , by [4, Equation (1.16)], it is given by µ K z ( x, λ ) := πx λ µ + z − sin( zπ ) ( (cid:16) x (cid:17) − z Γ( µ + λ + )Γ(1 − z )Γ( λ + − z ) F " µ + λ + λ + − z, − z ; x − (cid:16) x (cid:17) z Γ( µ + z + λ + )Γ(1 + z )Γ( λ + ) F " µ + z + λ + λ + , z ; x , (1.10)with µ K ( x, λ ) = lim z → µ K z ( x, λ ).From [4, equation (1.17)], we have − z K z ( x, λ ) = x λ K z ( x ) . Therefore, it is obvious to see that if µ = − z and λ = 0 then µ K z ( x, λ ) reduces to K z ( x ). Formore information on µ K z ( x, λ ) and its number theoretic applications, we refer the reader to[4].Our second generalization of the Watson’s result (1.3) is given below. Theorem 1.6.
Let
Re( x ) > , Re( z ) > and Re( µ + λ ) > . Let µ K z ( x, λ ) be defined in (1.10) . Then ∞ X n =1 (cid:16) nx (cid:17) z − λ µ K z ( nx, λ ) + 2 z + µ + λ − Γ( z )Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) + λ − z (cid:1) − √ π µ + z + λ Γ( λ + µ )Γ (cid:0) + z (cid:1) x Γ( λ − z )= 2 µ + λ − z √ π (cid:16) xπ (cid:17) z Γ (cid:0) z + (cid:1) Γ (cid:0) + λ + µ + z (cid:1) Γ (cid:0) + λ (cid:1) ∞ X n =1 n z +1 2 F " + z, + λ + µ + z + λ ; − x π n , (1.11) where F ( a, b ; c ; ξ ) is the Gauss hypergeometric function defined by [15, p. 110, Equation(5.4)] F ( a, b ; c ; ξ ) = Γ( c )Γ( b )Γ( c − b ) Z t b − (1 − t ) c − b − (1 − tξ ) − a dt, for Re( c ) > Re( b ) > , | arg(1 − ξ ) | < π . XTENSIONS OF WATSON’S THEOREM AND THE RAMANUJAN-GUINAND FORMULA 5
For Theorem 1.6 also, we remove the restriction on z in the following theorem. Theorem 1.7.
Let
Re( x ) > and Re( µ + λ ) > . Let M > be an integer. If Re( z ) > − M ,then ∞ X n =1 (cid:16) nx (cid:17) z − λ µ K z ( nx, λ ) + 2 z + µ + λ − Γ( z )Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) + λ − z (cid:1) − √ π µ + z + λ Γ( λ + µ )Γ (cid:0) z + (cid:1) x Γ( λ − z )= 2 µ + λ − z √ π (cid:16) xπ (cid:17) z Γ (cid:0) z + (cid:1) Γ (cid:0) + λ + µ + z (cid:1) Γ (cid:0) + λ (cid:1) ∞ X n =1 n z +1 ( F " + z, + λ + µ + z + λ ; − x π n − M − X m =0 (cid:0) + z (cid:1) m (cid:0) + λ + µ + z (cid:1) m m ! (cid:0) + λ (cid:1) m (cid:18) − x π n (cid:19) m ) + 2 µ + λ − z √ π (cid:16) xπ (cid:17) z M − X m =0 Γ (cid:0) z + + m (cid:1) Γ (cid:0) + λ + µ + z + m (cid:1) ζ (2 z + 2 m + 1) m !Γ (cid:0) + λ + m (cid:1) (cid:18) − x π (cid:19) m . (1.12)We separately record z = 0 case of the above theorem below. Theorem 1.8.
Let
Re( x ) > and Re( µ + λ ) > . We have ∞ X n =1 (cid:16) nx (cid:17) − λ µ K ( nx, λ ) − π µ + λ Γ( λ + µ ) x Γ( λ )= 2 µ + λ Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) + λ (cid:1) ∞ X n =1 n ( F " , + λ + µ + λ ; − x π n − ) + 2 µ + λ − Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) λ + (cid:1) (cid:18) (cid:16) γ − log (cid:16) x π (cid:17)(cid:17) − ψ (cid:18) λ + 12 (cid:19) + ψ (cid:18) λ + µ + 12 (cid:19)(cid:19) . (1.13)Theorem 1.8 gives a formula of Watson [18, p. 301, Equation (6)]. Corollary 1.9.
For
Re( x ) > , ∞ X n =1 K ( nx ) − πx = ∞ X n =1 (cid:26) π √ x + 4 n π − n (cid:27) + γ − log (cid:16) x π (cid:17) . This paper is organized as follows. Theorem 1.3 and Theorem 1.4 are proved in Section2. Section 3 is devoted to giving a new proof of Theorem 1.2. We derive Theorem 1.6, andTheorem 1.7 and its Corollary in Section 4 and its subsection 4.1.2.
Generalization of Watson’s result with K z,w ( x )The main ingredient to prove Theorem 1.3 and Theorem 1.6 is Poisson’s summation formulain the following form [16, p. 60-61]: Theorem 2.1. If f ( t ) is continuous and of bounded variation on (0 , ∞ ) , and if R ∞ f ( t ) dt exists, then f (0 + ) + 2 ∞ X n =1 f ( n ) = 2 Z ∞ f ( t ) dt + 4 ∞ X n =1 Z ∞ f ( t ) cos(2 πnt ) dt. RAHUL KUMAR
One of the nice properties of K z,w ( t ), derived in [5], is the following integral representationfor it [5, Theorem 1.7]. Let z, w ∈ C and | arg( x ) | < π , we have K z,w ( x ) = (cid:16) x (cid:17) − z Z ∞ e − u − x u cos( wu ) cos (cid:16) wxu (cid:17) u z − du. (2.1)To prove Theorem 1.3, it is imperative to obtain the following new integral evaluationwhich contain K z,w ( x ) in its integrand. Lemma 2.2.
Let w ∈ C , Re( z ) > , and | arg( x ) | < π . Let A ( n, z, w, x ) be defined in (1.6) .If a ≥ , then Z ∞ (cid:18) xt (cid:19) z K z,w ( xt ) cos( at ) dt = 14 √ π Γ (cid:18)
12 + z (cid:19) x z e − w ( x + a ) z + A (cid:16) a π , z, w, x (cid:17) . (2.2) Proof.
Invoke (2.1) to deduce that Z ∞ (cid:18) xt (cid:19) z K z,w ( xt ) cos( at ) dt = Z ∞ Z ∞ e − u − x t u cos( wu ) cos (cid:18) wxt u (cid:19) u z − cos( at ) dudt = Z ∞ e − u cos( wu ) u z − Z ∞ e − x t u cos (cid:18) wxt u (cid:19) cos( at ) dtdu, (2.3)where in the last step we interchanged the order of the integration which is justified becauseof the absolute convergence. Note that Z ∞ e − x t u cos (cid:18) wxt u (cid:19) cos( at ) dt = 12 Z ∞ e − x t u cos (cid:18) at + wxt u (cid:19) dt + 12 Z ∞ e − x t u cos (cid:18) at − wxt u (cid:19) dt (2.4)From [7, p. 488, Equation (3.896.4)], for Re( β ) >
0, we have Z ∞ e − βt cos ( bt ) dt = 12 r πβ exp (cid:18) − b β (cid:19) (2.5)Let β = x u and b = at + wx u in (2.5) to get Z ∞ e − x t u cos (cid:18) at + wxt u (cid:19) dt = √ πx u exp (cid:18) − (2 au + wx ) x (cid:19) . (2.6)Again use (2.5) with β = x u and b = at + = − wx u to find Z ∞ e − x t u cos (cid:18) at − wxt u (cid:19) dt = √ πx u exp (cid:18) − (2 au − wx ) x (cid:19) . (2.7)From (2.4), (2.6), and (2.7), Z ∞ e − x t u cos (cid:18) wxt u (cid:19) cos( at ) dt = √ π x u (cid:18) e − (2 au − wx )24 x + e − (2 au + wx )24 x (cid:19) = √ π x ue − w e − a u x (cid:16) e auwx + e − auwx (cid:17) = √ πx ue − w e − a u x cosh (cid:16) auwx (cid:17) . (2.8) XTENSIONS OF WATSON’S THEOREM AND THE RAMANUJAN-GUINAND FORMULA 7
Substitute value from (2.8) in (2.3) so as to obtain Z ∞ (cid:18) xt (cid:19) z K z,w ( xt ) cos( at ) dt = √ πx e − w Z ∞ e − (cid:16) a x +1 (cid:17) u cos( wu ) cosh (cid:16) auwx (cid:17) u z du = √ π x e − w ( Z ∞ e − (cid:16) a x +1 (cid:17) u u z cos (cid:18) wu + iauwx (cid:19) du + Z ∞ e − (cid:16) a x +1 (cid:17) u u z cos (cid:18) wu − iauwx (cid:19) du ) , (2.9)where in the last step we used the elementary fact 2 cos( A ) cos( B ) = cos( A + B ) + cos( A − B ).From [7, p. 503, Equation (3.952.8)] , for Re( β ) >
0, Re( µ ) > ν ∈ C , we have Z ∞ u µ − e − βu cos( νu ) du = 12 β − µ Γ (cid:16) µ (cid:17) e − ν β F (cid:18) − µ ν β (cid:19) . (2.10)Let µ = 2 z + 1 , β = a x + 1 and ν = wu + iauwx in (2.10) to get Z ∞ e − (cid:16) a x +1 (cid:17) u u z cos (cid:18) wu + iauwx (cid:19) du = 12 Γ (cid:18)
12 + z (cid:19) x z ( x + a ) z + F (cid:18) z + 12 ; 12 ; w ( a − ix )4( a + ix ) (cid:19) . (2.11)Now let µ = 2 z + 1 , β = a x + 1 and ν = wu − iauwx in (2.10) so that Z ∞ e − (cid:16) a x +1 (cid:17) u u z cos (cid:18) wu − iauwx (cid:19) du = 12 Γ (cid:18)
12 + z (cid:19) x z ( x + a ) z + F (cid:18) z + 12 ; 12 ; w ( a + ix )4( a − ix ) (cid:19) . (2.12)Finally substitute values from (2.11) and (2.12) in (2.9) and use (1.6) to arrive at (2.2). (cid:3) As a special case of Lemma 2.2, we get [18, p. 299].
Corollary 2.3.
Let z ∈ C and | arg( x ) | < π . If a ≥ , then Z ∞ (cid:18) xt (cid:19) z K z ( xt ) cos( at ) dt = 12 √ π Γ (cid:18)
12 + z (cid:19) x z ( x + a ) z + . (2.13) Proof.
Let w = 0 in Lemma 2.2 and use the fact that K z, ( xt ) = K z ( xt ) and F ( b ; c ; 0) = 1to get (2.13). (cid:3) We now have all necessary results to prove Theorem 1.3.
Proof of Theorem . Let f ( t ) = (cid:0) xt (cid:1) z K z,w ( xt ) in Theorem 2.1. From [5, Theorem 1.13(i)],as x → K z,w ( x ) ∼
12 Γ( z ) (cid:16) x (cid:17) − z F (cid:18) z ; 12 ; − w (cid:19) . (2.14)From [5, Theorem 1.12], we known that (cid:0) xt (cid:1) z K z,w ( xt ) has exponential decay, therefore, alongwith (2.14) it easy to see that the integral R ∞ f ( t ) dt converges. Now by using (2.14), we see Condition on ν is given ν >
0, however, it is easy to see that this result is actually true for all ν ∈ C . RAHUL KUMAR that lim x → (cid:18)(cid:18) xt (cid:19) z K z,w ( xt ) (cid:19) = f (0 + ) = 12 Γ( z ) F (cid:18) z ; 12 ; − w (cid:19) . (2.15)Let a = 0 in Lemma 2.2 to find Z ∞ f ( t ) dt = 12 x √ π Γ (cid:18)
12 + z (cid:19) e − w F (cid:18) z + 12 ; 12 ; − w (cid:19) . (2.16)Again invoke Theorem 2.2 with a = 2 πn so that Z ∞ f ( t ) cos(2 πnt ) dt = 14 √ π Γ (cid:18)
12 + z (cid:19) x z e − w ( x + 4 π n ) z + A ( n, z, w, x ) . (2.17)Substitute values from (2.15), (2.16) and (2.17) in Theorem 2.1 to establish (1.7). (cid:3) Analytic continuation of Theorem 1.3.
This subsection is dedicated to findinganalytic continuation of Theorem 1.3, that is, to prove Theorem 1.4.
Proof of Theorem . Note that, for large n ,1( x + 4 n π ) z + = 1(2 nπ ) z +1 (cid:18) x n π (cid:19) − z − = 1(2 nπ ) z +1 ∞ X m =0 (cid:18) − z − m (cid:19) (cid:18) x n π (cid:19) m = 1(2 nπ ) z +1 M − X m =0 (cid:18) − z − m (cid:19) (cid:18) x n π (cid:19) m + O (cid:18) n z +2 M +1 (cid:19) . (2.18)Now observe that ∞ X n =1 A ( n, z, w, x )( x + 4 n π ) z + = ∞ X n =1 A ( n, z, w, x ) ( x + 4 n π ) z + − nπ ) z +1 M − X m =0 (cid:18) − z − m (cid:19) × (cid:18) x n π (cid:19) m + 1(2 nπ ) z +1 M − X m =0 (cid:18) − z − m (cid:19) (cid:18) x n π (cid:19) m ) = ∞ X n =1 A ( n, z, w, x ) ( x + 4 n π ) z + − M − X m =0 (cid:18) − z − m (cid:19) x m (2 nπ ) z +2 m +1 ) + ∞ X n =1 A ( n, z, w, x ) M − X m =0 (cid:18) − z − m (cid:19) x m (2 nπ ) z +2 m +1 = ∞ X n =1 A ( n, z, w, x ) ( x + 4 n π ) z + − M − X m =0 (cid:18) − z − m (cid:19) x m (2 nπ ) z +2 m +1 ) + M − X m =0 (cid:18) − z − m (cid:19) x m (2 π ) z +2 m +1 ∞ X n =1 A ( n, z, w, x ) n z +2 m +1 . (2.19) XTENSIONS OF WATSON’S THEOREM AND THE RAMANUJAN-GUINAND FORMULA 9
Substitute (2.19) in (1.3) to obtain2 ∞ X n =1 (cid:16) nx (cid:17) z K z,w ( nx ) + 12 Γ( z ) F (cid:18) z ; 12 ; − w (cid:19) − √ πx Γ (cid:18)
12 + z (cid:19) e − w F (cid:18)
12 + z ; 12 ; − w (cid:19) = √ πx z Γ (cid:18)
12 + z (cid:19) e − w " ∞ X n =1 A ( n, z, w, x ) ( x + 4 n π ) z + − M − X m =0 (cid:18) − z − m (cid:19) x m (2 nπ ) z +2 m +1 ) + M − X m =0 (cid:18) − z − m (cid:19) x m (2 π ) z +2 m +1 ∞ X n =1 A ( n, z, w, x ) n z +2 m +1 . (2.20)By employing the asymptotic expansion of K z,w ( x ), given in [5, Theorem 1.12], it is easyto see that as the series on the right-hand side of (2.20) converges uniformly as a functionof z and its summand is also analytic, therefore, by using Weierstrass’ theorem on analyticfunction it represents an analytic function in Re( z ) > − M . Hence the right-hand side of(2.20) is analytic in z in Re( z ) > − M .By using (2.18) and the fact A ( n, z, w, x ) = O (1), for large n , we get A ( n, z, w, x ) ( x + 4 n π ) z + − M − X m =0 (cid:18) − z − m (cid:19) x m (2 nπ ) z +2 m +1 ) = O (cid:18) n z +2 M +1 (cid:19) . (2.21)Upon employing (2.21) is it is easy to see that the infinite series on the left-hand side of(2.20) is uniformly convergent as a function of z in Re( z ) > − M . The summand of this seriesis also analytic in this region. Therefore, the left-hand side of (2.20) represents an analyticfunction of z in Re( z ) > − M . Now results follows by the principle of analytic continuationfor the conditions imposed in the hypotheses of the theorem. (cid:3) Proof of Corollary . Let w = 0 in Theorem 1.4. Use the fact K z, ( x ) = K z ( x ) and A ( n, z, , x ) = 2 to arrive at (1.9). (cid:3) A new proof of Dixit-Kesarwani-Moll’s generalization of theRamanujan-Guinand formula
In this section, as an application of our Theorem 1.3, we provide a new proof of Dixit-Kesarwani-Moll’s generalization of the Ramanujan-Guinand formula, that is, Theorem 1.2.
Proof of Theorem . Using the definition of the divisor function σ − z ( n ) = P d | n d − z , notethat √ α ∞ X n =1 σ − z ( n ) n z e − w K z ,iw (2 nα ) = √ αe − w ∞ X n =1 X d | n d − z n z K z ,iw (2 nα )= √ αe − w ∞ X d =1 d − z ∞ X k =1 k z K z ,iw (2 kdα ) . (3.1)Let x = 2 dα and replace w by iw in Theorem 1.3 to see ∞ X n =1 n z K z ,iw (2 ndα ) = − α − z d z Γ (cid:16) z (cid:17) F (cid:18) z w (cid:19) + 14 √ πe w α − z − d z +1 Γ (cid:18) z + 12 (cid:19) × F (cid:18) z w (cid:19) + √ π − z α z d − z Γ (cid:18) z (cid:19) e w ∞ X n =1 A (cid:0) n, z , iw, dα (cid:1) (4 d α + 4 π n ) z +12 . (3.2)Substitute value from (3.2) in (3.1) to conclude √ α ∞ X n =1 σ − z ( n ) n z e − w K z ,iw (2 nα ) = √ αe − w ( − α − z Γ (cid:16) z (cid:17) F (cid:18) z w (cid:19) ∞ X d =1 d z + 14 √ πe w α − z − Γ (cid:18) z + 12 (cid:19) F (cid:18) z w (cid:19) ∞ X d =1 d z +1 + 14 √ πe w α z Γ (cid:18) z (cid:19) ∞ X d =1 ∞ X n =1 A (cid:0) n, z , iw, dα (cid:1) ( d α + π n ) z +12 ) . Use the definition of the Riemann zeta function ζ ( z ) in the above equation to get √ α ∞ X n =1 σ − z ( n ) n z e − w K z ,iw (2 nα ) = − α − z Γ (cid:16) z (cid:17) ζ ( z ) e − w F (cid:18) z w (cid:19) + 14 √ πα − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) F (cid:18) z w (cid:19) + 14 √ πα z +12 Γ (cid:18) z (cid:19) ∞ X d =1 ∞ X n =1 A (cid:0) n, z , iw, dα (cid:1) ( d α + π n ) z +12 . (3.3)Upon using the fact αβ = π in the last term of (3.3), we find that √ α ∞ X n =1 σ − z ( n ) n z e − w K z ,iw (2 nα ) = − α − z Γ (cid:16) z (cid:17) ζ ( z ) e − w F (cid:18) z w (cid:19) + 14 √ πα − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) F (cid:18) z w (cid:19) + 14 √ πβ z +12 Γ (cid:18) z (cid:19) ∞ X d =1 ∞ X n =1 A (cid:0) n, z , iw, dα (cid:1) ( d π + n β ) z +12 . (3.4)Replace α by β and w by iw in (3.3) so that p β ∞ X n =1 σ − z ( n ) n z e w K z ,w (2 nβ ) = − β − z Γ (cid:16) z (cid:17) ζ ( z ) e w F (cid:18) z − w (cid:19) + 14 √ πβ − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) F (cid:18) z − w (cid:19) + 14 √ πβ z +12 Γ (cid:18) z (cid:19) ∞ X d =1 ∞ X n =1 A (cid:0) n, z , w, dβ (cid:1) ( n π + d β ) z +12 , (3.5)where we used the facts that K z ,w (2 nβ ) and A (cid:0) n, z , w, dβ (cid:1) are even functions of w . Nowinterchange the role of the variables n and d in the last term of (3.5) and then interchange XTENSIONS OF WATSON’S THEOREM AND THE RAMANUJAN-GUINAND FORMULA 11 the order of summation to get p β ∞ X n =1 σ − z ( n ) n z e w K z ,w (2 nβ ) = − β − z Γ (cid:16) z (cid:17) ζ ( z ) e w F (cid:18) z − w (cid:19) + 14 √ πβ − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) F (cid:18) z − w (cid:19) + 14 √ πβ z +12 Γ (cid:18) z (cid:19) ∞ X d =1 ∞ X n =1 A (cid:0) d, z , w, nβ (cid:1) ( d π + n β ) z +12 . (3.6)From the definition (1.6) of A (cid:0) n, z , iw, dα (cid:1) and the fact αβ = π , we have A (cid:16) n, z , iw, dα (cid:17) = F (cid:18)
12 + z ; 12 ; − w (2 nπ + 2 idα )8 nπ − idα (cid:19) + F (cid:18)
12 + z ; 12 ; − w (2 nπ − idα )8 nπ + 8 idα (cid:19) = F (cid:18)
12 + z ; 12 ; − w (2 nπ + 2 idπ /β )8 nπ − idπ /β (cid:19) + F (cid:18)
12 + z ; 12 ; − w (2 nπ − idπ /β )8 nπ + 8 idπ /β (cid:19) = F (cid:18)
12 + z ; 12 ; − w (2 nβ + 2 idπ )8 nβ − idπ (cid:19) + F (cid:18)
12 + z ; 12 ; − w (2 nβ − idπ )8 nβ + 8 idπ (cid:19) = F (cid:18)
12 + z ; 12 ; − w (2 nβi − dπ )8 nβi + 8 dπ (cid:19) + F (cid:18)
12 + z ; 12 ; − w (2 nβi + 2 dπ )8 nβi − dπ (cid:19) = F (cid:18)
12 + z ; 12 ; w (2 dπ − nβi )8 nβi + 8 dπ (cid:19) + F (cid:18)
12 + z ; 12 ; w (2 nβi + 2 dπ )8 dπ − nβi (cid:19) = A (cid:16) d, z , w, nβ (cid:17) . (3.7)Employ (3.7) in (3.6) and then subtract the resulting expression from (3.4) and simplify toarrive at √ α ∞ X n =1 σ − z ( n ) n z e − w K z ,iw (2 nα ) − p β ∞ X n =1 σ − z ( n ) n z e w K z ,w (2 nβ )= − α − z Γ (cid:16) z (cid:17) ζ ( z ) e − w F (cid:18) z w (cid:19) + 14 √ πα − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) F (cid:18) z w (cid:19) + 14 β − z Γ (cid:16) z (cid:17) ζ ( z ) e w F (cid:18) z − w (cid:19) − √ πβ − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) F (cid:18) z − w (cid:19) . (3.8)We have Kummer’s transformation [15, P. 173, Equation (7.5)] F ( b ; c ; z ) = e z F ( c − b ; c ; − z ) . (3.9)Use (3.9) twice in (3.8) to find √ α ∞ X n =1 σ − z ( n ) n z e − w K z ,iw (2 nα ) − p β ∞ X n =1 σ − z ( n ) n z e w K z ,w (2 nβ )= − α − z Γ (cid:16) z (cid:17) ζ ( z ) F (cid:18) − z − w (cid:19) + 14 √ πα − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) F (cid:18) z w (cid:19) + 14 β − z Γ (cid:16) z (cid:17) ζ ( z ) F (cid:18) z w (cid:19) − √ πβ − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) F (cid:18) z − w (cid:19) . (3.10) Now invoke the functional equation of ζ ( s ) [17, p.22, Equation (2.6.40] π − s Γ (cid:16) s (cid:17) ζ ( s ) = π − − s Γ (cid:18) − s (cid:19) ζ (1 − s ) , along with letting s = z + 1 and use the hypothesis αβ = π so that14 √ πα − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) F (cid:18) z w (cid:19) − √ πβ − z − Γ (cid:18) z + 12 (cid:19) ζ ( z + 1) × F (cid:18) z − w (cid:19) = 14 β z +12 Γ (cid:16) − z (cid:17) ζ ( − z ) F (cid:18) z w (cid:19) − α z +12 Γ (cid:16) − z (cid:17) ζ ( − z ) F (cid:18) z − w (cid:19) . Substitute the above value in (3.10) so as to deduce that √ α ∞ X n =1 σ − z ( n ) n z e − w K z ,iw (2 nα ) − p β ∞ X n =1 σ − z ( n ) n z e w K z ,w (2 nβ )= − α − z Γ (cid:16) z (cid:17) ζ ( z ) F (cid:18) − z − w (cid:19) + 14 β z +12 Γ (cid:16) − z (cid:17) ζ ( − z ) F (cid:18) z w (cid:19) + 14 β − z Γ (cid:16) z (cid:17) ζ ( z ) F (cid:18) z w (cid:19) − α z +12 Γ (cid:16) − z (cid:17) ζ ( − z ) F (cid:18) z − w (cid:19) . Now simplify the above equation to prove (1.5). (cid:3) Generalization of Watson’s result in the setting of µ K z ( x, λ )This section is devoted to proving Theorem 1.6. However, to prove it we need to evaluatethe following integral involving the generalized modified Bessel function µ K z ( x, λ ). Lemma 4.1.
Let
Re( z ) > and x, µ, λ ∈ C such that Re( µ + λ ) > . Then for a ≥ , wehave Z ∞ (cid:18) tx (cid:19) z − λ µ K z ( tx, λ ) cos( at ) dt = √ π µ + z + λ − x z a z +1 Γ (cid:0) + z (cid:1) Γ (cid:0) + µ + z + λ (cid:1) Γ (cid:0) + λ (cid:1) F " + z, + λ + µ + z + λ ; − x a . Proof.
From (1.10) employ the definition of µ K z ( x, λ ), Z ∞ (cid:18) tx (cid:19) z − λ µ K z ( tx, λ ) cos( at ) dt = π µ + λ + z − Γ (cid:0) µ + λ + (cid:1) Γ (cid:0) + λ − z (cid:1) Γ (1 − z ) Z ∞ cos( at ) F (cid:18) µ + λ + 12 ; 12 + λ − z ; 1 − z ; ( tx ) (cid:19) dt − πx z µ + λ − z − Γ (cid:0) µ + z + λ + (cid:1) sin( πz )Γ(1 + z )Γ (cid:0) λ + (cid:1) Z ∞ t z cos( at ) F (cid:18) µ + z + λ + 12 ; λ + 12 , z ; ( tx ) (cid:19) dt. (4.1) XTENSIONS OF WATSON’S THEOREM AND THE RAMANUJAN-GUINAND FORMULA 13
From [7, p. 818, Equation (7.542.3)] , for y ≥ , p ≤ q − σ ) > | Re( ν ) | , we have Z ∞ t σ − p F q ( a , ..., a p ; b , ..., b q ; − ηt ) Y ν ( yt ) dt = − σ − π y − σ cos (cid:16) π σ − ν ) (cid:17) Γ (cid:18) σ + ν (cid:19) Γ (cid:18) σ − ν (cid:19) p +2 F q (cid:18) a , ..., a p , σ + ν , σ − ν b , ..., b q ; 4 ηy (cid:19) . (4.2)Let p = 1 , q = 2 , ν = in (4.2) and the fact that Y ( x ) = − p π x cos( x ) to find Z ∞ t σ − F ( a ; b , b ; − ηt ) cos( yt ) dt = 2 σ − √ π y − σ cos (cid:18) π (cid:18) σ − (cid:19)(cid:19) Γ (cid:18) σ + 1 / (cid:19) Γ (cid:18) σ − / (cid:19) × p +2 F q (cid:18) a , σ + 1 / , σ − /
22 ; b , b ; 4 ηy (cid:19) . (4.3)Let σ = , y = a and a = µ + λ + , b = λ + − z, b = 1 − z and η = − x in (4.3), thennote that cosine term on the right-hand side vanishes; therefore, we get Z ∞ F (cid:18) µ + λ + 12 ; λ + 12 − z, − z ; x t (cid:19) cos( at ) dt = 0 (4.4)Now let σ = + 2 z, y = a and a = µ + z + λ + , b = λ + , b = 1 + z and η = − x in(4.3) to see that Z ∞ t z F (cid:18) µ + z + λ + 12 ; λ + 12 , z ; x t (cid:19) cos( at ) dt = − Γ(2 z + 1) sin( πz ) a z +1 2 F (cid:18)
12 + z,
12 + λ + µ + z ; 12 + λ ; − x a (cid:19) . (4.5)Lemma 4.1 now follows from (4.1), (4.4) and (4.5). (cid:3) We are now ready to prove Theorem 1.6.
Proof of Theorem . Let f ( t ) := (cid:18) tx (cid:19) z − λ µ K z ( tx, λ ) . (4.6)From (1.10) and (4.6), f ( t ) = π λ + µ + z − sin( zπ ) (cid:26) Γ( µ + λ + )Γ(1 − z )Γ( λ + − z ) F " µ + λ + λ + − z, − z ; ( xt ) − (cid:18) xt (cid:19) z Γ( µ + z + λ + )Γ(1 + z )Γ( λ + ) F " µ + z + λ + λ + , z ; ( xt ) . (4.7)It is easy to see from the above equation that as t →
0, for Re( z ) > f ( t ) = O (1) . (4.8) minus sign in the argument of the p F q on the right-hand side must replaced by the plus sign From [4, Lemma 7.1], as t → ∞ , f ( t ) = O (cid:16) t − λ − µ − (cid:17) . (4.9)Now by using (4.8) and (4.9), it is clear that the integral R ∞ f ( t ) dt exists for Re( z ) > µ + λ ) > t = 0 and use Γ( z )Γ(1 − z ) = π/ sin( πz ) in (4.7) so that, for Re( z ) > f (0) = 2 λ + µ + z − Γ( µ + λ + )Γ( z )Γ( λ + − z ) . (4.10)Employ Lemma 4.1 with a = 0 and (4.6) to get Z ∞ f ( t ) dt = √ π µ + z + λ − Γ( µ + λ )Γ (cid:0) + z (cid:1) x Γ( λ − z ) . (4.11)Again upon invoking Lemma 4.1 with a = 2 πn , we have Z ∞ f ( t ) cos(2 πnt ) dt = √ π µ + z + λ − x z (2 πn ) z +1 Γ (cid:18)
12 + z (cid:19) Γ (cid:0) + µ + z + λ (cid:1) Γ (cid:0) + λ (cid:1) × F (cid:18)
12 + z,
12 + µ + z + λ ; 12 + λ ; − x π n (cid:19) . (4.12)Substitute values from (4.6), (4.10), (4.11) and (4.12) in Theorem 2.1 and simplify to arriveat (1.11). (cid:3) Analytic continuation of Theorem 1.6.
Proof of Theorem . Upon using the series definition of F , we see that as n → ∞ , F " + z, + λ + µ + z + λ ; − x π n = M X m =0 (cid:0) + z (cid:1) m (cid:0) + λ + µ + z (cid:1) m m ! (cid:0) + λ (cid:1) m (cid:18) − x π n (cid:19) − m + O (cid:0) n − M (cid:1) . (4.13)Add and subtract P Mm =0 ( + z ) m ( + λ + µ + z ) m m ! ( + λ ) m (cid:16) − x π n (cid:17) − m in the first step below to see that ∞ X n =1 n z +1 2 F " + z, + λ + µ + z + λ ; − x π n = ∞ X n =1 n z +1 ( F " + z, + λ + µ + z + λ ; − x π n − M − X m =0 (cid:0) + z (cid:1) m (cid:0) + λ + µ + z (cid:1) m m ! (cid:0) + λ (cid:1) m × (cid:18) − x π n (cid:19) m ) + ∞ X n =1 n z +1 M − X m =0 (cid:0) + z (cid:1) m (cid:0) + λ + µ + z (cid:1) m m ! (cid:0) + λ (cid:1) m (cid:18) − x π n (cid:19) m = ∞ X n =1 n z +1 ( F " + z, + λ + µ + z + λ ; − x π n − M − X m =0 (cid:0) + z (cid:1) m (cid:0) + λ + µ + z (cid:1) m m ! (cid:0) + λ (cid:1) m × (cid:18) − x π n (cid:19) m ) + M − X m =0 (cid:0) + z (cid:1) m (cid:0) + λ + µ + z (cid:1) m ζ (2 z + 2 m + 1) m ! (cid:0) + λ (cid:1) m (cid:18) − x π (cid:19) m . (4.14) XTENSIONS OF WATSON’S THEOREM AND THE RAMANUJAN-GUINAND FORMULA 15
Employ (4.14) in (1.11) to deduce that2 ∞ X n =1 (cid:16) nx (cid:17) z − λ µ K z ( nx, λ ) + 2 z + µ + λ − Γ( z )Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) + λ − z (cid:1) − √ π µ + z + λ Γ( λ + µ )Γ (cid:0) + z (cid:1) x Γ( λ − z )= 2 µ + λ − z √ π (cid:16) xπ (cid:17) z Γ (cid:0) z + (cid:1) Γ (cid:0) + λ + µ + z (cid:1) Γ (cid:0) + λ (cid:1) ∞ X n =1 n z +1 ( F " + z, + λ + µ + z + λ ; − x π n − M − X m =0 (cid:0) + z (cid:1) m (cid:0) + λ + µ + z (cid:1) m m ! (cid:0) + λ (cid:1) m (cid:18) − x π n (cid:19) m ) + 2 µ + λ − z √ π (cid:16) xπ (cid:17) z M − X m =0 Γ (cid:0) z + + m (cid:1) Γ (cid:0) + λ + µ + z + m (cid:1) ζ (2 z + 2 m + 1) m !Γ (cid:0) + λ + m (cid:1) (cid:18) − x π (cid:19) m . (4.15)By invoking [4, Lemma 7.1], it is easy to see that the series on the left-hand side of (4.15) isuniformly convergent Re( z ) > − M . Also, the summand of this series is an analytic function,therefore, this series represents an analytic function of z in Re( z ) > − M by Weierstrass’theorem for analytic functions.By using (4.13) observe that the summand of the infinite series on the right-hand side of(4.15) is O (cid:0) n − z − m − (cid:1) when n is large. Therefore this series converges uniformly as functionof z in Re( z ) > − M . Since the summand of the infinite series on the right-hand side is analyticin Re( z ) > − M , we see by Weierstrass’ theorem that this series represents an analytic functionof z for Re( z ) > − M . Therefore, by the principle of analytic continuation, we see that (4.15)holds for Re( z ) > − M with having limiting values on the poles at 0 , − , − , − , ..., − M + and Re( x ) > µ + λ ) >
0. This proves the theorem. (cid:3)
Theorem 1.8 is proved next.
Proof of Theorem . Let M = 1 in Theorem 1.7. Let z → ∞ X n =1 (cid:16) nx (cid:17) − λ µ K ( nx, λ ) − π µ + λ Γ( λ + µ ) x Γ( λ )= 2 µ + λ Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) + λ (cid:1) ∞ X n =1 n ( F " , + λ + µ + λ ; − x π n − ) + lim z → ( µ + λ − z √ π (cid:16) xπ (cid:17) z Γ (cid:0) z + (cid:1) Γ (cid:0) + λ + µ + z (cid:1) ζ (2 z + 1)Γ (cid:0) + λ (cid:1) − z + µ + λ − Γ( z )Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) + λ − z (cid:1) ) . (4.16)We next evaluate the limit in the above equation. We have the following well-known expan-sions, as z →
0, 2 − z = 1 − log(2) z + O ( z ) , (cid:16) x π (cid:17) z = 1 + 2 z log (cid:16) xπ (cid:17) + O ( z ) , Γ (cid:18)
12 + z (cid:19) = √ π (cid:18) ψ (cid:18) (cid:19) z (cid:19) + O ( z ) , Γ (cid:18)
12 + λ + µ + z (cid:19) = Γ (cid:18)
12 + λ + µ (cid:19) (cid:18) ψ (cid:18)
12 + λ + µ (cid:19) z (cid:19) + O ( z ) , ζ (2 z + 1) = 12 z + γ + O ( z ) . Also, 2 z = 1 + log(2) z + O ( z ) , Γ( z ) = 1 z − γ + O ( z ) , (cid:0) + λ − z (cid:1) = 1Γ (cid:0) + λ (cid:1) (cid:18) ψ (cid:18)
12 + λ (cid:19) z (cid:19) + O ( z ) . Use all these expansion to deduce thatlim z → ( µ + λ − z √ π (cid:16) xπ (cid:17) z Γ (cid:0) z + (cid:1) Γ (cid:0) + λ + µ + z (cid:1) ζ (2 z + 1)Γ (cid:0) + λ (cid:1) − z + µ + λ − Γ( z )Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) + λ − z (cid:1) ) = 2 µ + λ − Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) + λ (cid:1) (cid:18) x ) − π ) + 3 γ + ψ (cid:18) (cid:19) + ψ (cid:18)
12 + λ + µ (cid:19) − ψ (cid:18)
12 + λ (cid:19)(cid:19) , = 2 µ + λ − Γ (cid:0) + λ + µ (cid:1) Γ (cid:0) + λ (cid:1) (cid:18) x ) − π ) + 2 γ + ψ (cid:18)
12 + λ + µ (cid:19) − ψ (cid:18)
12 + λ (cid:19)(cid:19) , (4.17)where we used the fact ψ (cid:0) (cid:1) = − − γ . Now substitute the limit evaluation (4.17) in(4.16) to get (1.13). (cid:3) Proof of Corollary . Let µ = 0 in Theorem 1.8 and let λ → K ( x,
0) = K ( x ) and then use F (cid:0) ; − ; − x (cid:1) = √ x to get the result. (cid:3) Acknowledgements
The author sincerely thanks Professor Atul Dixit for a careful reading of this article, hisvaluable suggestions and his support throughout this work. The author is an institute post-doctoral fellow at IIT Gandhinagar and sincerely thanks the institute and Professor AtulDixit for their financial support.
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