Integral representations for the Dirichlet L-functions and their expansions in Meixner-Pollaczek polynomials and rising factorials A. Kuznetsov Dept. of Mathematical Sciences University of New Brunswick Saint John, NB EL L5, Canada e-mail: akuznets@unbsj.ca Current version: April, 7 Abstract In this article we provide integral representations for the Dirichlet beta and Riemann zeta functions, which are obtained by combining Mellin transform with the fractional Fourier transform. As an application of these integral formulas we derive tractable expansions of these L-functions in the series of Meixner-Pollaczek polynomials and rising factorials. Keywords: Riemann zeta function, Meixner-Pollaczek polynomials, rising factorials, confluent hypergeometric function, Mellin transform, fractional Fourier transform This is the author s version of the work. It is posted here by permission of Taylor& Francis for personal use, not for redistribution. The definitive version was published in Integral Transforms and Special Functions, Volume, Issue, January 7. doi:./65675773
Introduction In this article we study the relations between Mellin transform and the fractional powers of Fourier transform and we show how these integral transforms can be used to obtain new results for the Dirichlet L-functions. To illustrate the ideas let us consider the Dirichlet beta function βs, which for Res > is defined as βs = n. This function can be obtained as the Mellin n+ s transform of fx = sech x, and the functional equation for βs follows from the fact that fx is invariant under cosine transform F c and the integral kernel x s of the Mellin transform is also invariant under F c, with s replaced by s and some multiplication by functions of s only. However, more can be said about function fx: not only it is invariant under F c, but infinitely many fractional powers F m n c f y have quite simple form. But since the integral kernel of the Mellin transform is not invariant under fractional powers of F c in fact we obtain a transform with the confluent hypergeometric function as an integral kernel, we obtain new integral representations for βs. These integral representations Theorems. and 5. provide a simple way of obtaining tractable expansions of the Dirichlet L-functions in the series of polynomials, such as Meixner- Pollaczek polynomials or rising factorials. The partial sums of these expansions also satisfy functional equation and the coefficients have a very simple form of the exponential generating function. In a companion paper 9] we study in more detail the expansion of the Riemann Ξ-function in Meixner-Pollaczek polynomials and the zeros of the partial sums of this expansion see also ], 5], 6] and 7]. This article is organized as follows: in section we provide some background material on the fractional Fourier transform and its relation to the Mellin transform. Section 3 is devoted to Mordell integrals, which play the central role in this article since they allow us to compute the fractional Fourier transform of certain hyperbolic functions. In section we study Dirichlet beta function, derive integral representation and expansions in Meixner-Pollaczek polynomials and in rising factorials. Only then in section 5 we study the famous Riemann zeta function ζs: it wasn t given a priority because in this case analysis is somewhat more complicated by the presence of the poles. At last we would like to mention that everywhere in this article we use the following notation for the rising factorial: a n = aa +... a + n. Fractional Fourier and scaled Mellin transforms In this section we review some background material on the fractional Fourier sine cosine transform and Mellin transform see ], 7] which will be used extensively later. The results are presented in the framework of the fractional Hankel transform H a ν, which includes both fractional cosine and sine transforms as the special cases ν = ±. Let us define A = L,, dx to be Hilbert space of the square integrable functions on,. We choose a complete orthogonal basis φ n x = L ν nx e x x ν+ ν >, where L ν nx are Laguerre polynomials see ], ]. The fractional Hankel transform H a ν of order a is a unitary
operator on A defined by H a νφ n = e ina φ n see 7]. Below we present some of the properties of the fractional Hankel transform which will be used later: H a ν is a unitary operator on A such that H a νh b ν = H a+b ν The integral kernel and H a ν = H a ν = H a ν. e x +y xy ν+ n! Γn + ν + e ina L ν nx L ν ny = = e i ν+a â sin a e i where â = signsin a see 7]. cot a x +y xyjν xy sin a When ν = we obtain the fractional cosine transform F a c with the integral kernel e i a â sin e i a cot x +y xy cos a sin. a When ν = we obtain the fractional sine transform F a s with the integral kernel e 3i a â sin e i a cot x +y xy sin a sin. 3 a Next we define another Hilbert space B = L R, Γ λ + i t dt. Here and everywhere else in this article λ = ν+ note that λ >. The scaled Mellin transform is defined as, M λ fs = s Γ λ + s Mfs, where Mfs = fxx s dx is the classical Mellin transform. The following properties of the scaled Mellin transform will be used later: M λ φ n s = λ i n P n λ t, where s = λ + it and P t n are the Meixner-Pollaczek polynomials see ] Parseval identity: if gt = M λ fs, s = + it, then f A = g B. The action of M λ on Fourier cosine F c = H M F c fs = M f s, M 3 and Fourier sine transforms F c = H : F s fs = M 3 f s. 3
Next we present a result which will be our main tool in the following sections: it describes the action of the scaled Mellin transform M λ on the fractional Hankel transform H ν λ = ν+. Proposition.. Assume fx A and let gt = M λ H ν ft. Then gt B, f A = g B and gt can be represented as gt = e t Γλ x λ e i x F λ + i t, λ; ix fxdx, Furthermore, fx = H ν M gx can be expressed as the following integral λ fx = xλ e i x Γλ R F λ + i t, λ; ix e t gt Γ λ + i t dt. Proof: The integral kernel of transformation M λ H ν is given by = i e λ s Γ λ + s y s e i x +y xyj ν xy dy = Γλ e t x λ e i x F λ + i t, λ; ix. Similarly, the integral kernel of the inverse transformation H ν M λ is given by i e λ + s Γ λ + s y s e i x +y xyj ν xy dy = = Γλ e t x λ e i x F λ + i t, λ; ix Γ λ + i t. Both integrals were computed with the help of ]. 3 Mordell integrals In this section we review several results about function e i τx cosxy hy, τ = cosh dx, Imτ. x Integrals of this type were used by Riemann to obtain functional equation and asymptotic formula for zeta function see 5]. Later these integrals were studied by Ramanujan 3] and by Mordell,
who analyzed their behavior with respect to modular transformations see ], ]. An extensive collection of facts about function hy, τ can be found in 6]. In the derivation of the integral representations for Dirichlet L-functions we will need explicit formulas for hy, τ, which can be obtained using the following functional equations see 6] for the proof hy, τ + hy + i, τ = τ e i i τ y+i, 5 hy, τ + e y iτ hy + i τ, τ = e iτ y. 6 If τ = m is an irreducible fraction, then by iterating Eq. 5 m times and Eq. 6 n times we n obtain a system of two linear equations in two variables h = hy, τ and h = hy + i m, τ. After eliminating h from these equations and simplifying the resulting formula for h we obtain hy, τ = cosh n y G, y, τ, n + e i i τ y G y τ,,, m], 7 τ τ where the quadratic Gauss sum is defined as G a,± y, τ, n = n ± k e iτk+a +n k ay. The following two integrals can also be expressed in terms of hy, τ and thus computed explicitly for rational τ: e i τx sin xy e x + dx = i τ e i τ y + Φ e i y τ e n y + n e n y k= G,+ y, τ, n e i i τ y G y,,, m] τ τ τ + e i τx sin xy e x dx = i τ e i τ y + Φ e i y τ + 9 ] e n y + n+m e n y G,+ y, τ, n e i i τ y G y,+,, m τ τ τ Dirichlet beta function In this section we illustrate the interplay between Mellin transform and the fractional cosine transform on the example of the Dirichlet beta function βs, which for Res > can be defined as βs = L s, χ = n. We also define ξ s, χ n+ s = s s Γ +s βs and Ξ t, χ = ξ + it, χ. Our main result in this section is the following integral representation for Ξt, χ : 5
Theorem.. For all t C y sin + Ξt, χ e t = e i y F + i t, ; iy cosh dy. y Proof: We start with the function fx = sech x and find that M fs = ξs, χ, Res >. Note that the functional equation ξs, χ = ξ s, χ follows at once from Eq. and the fact that fx is invariant under Fourier cosine transform F c see ]. Now we use Eq. 7 and compute F c fy: F y = F c fy = C, e i x +y cos xy cosh sin xdx = 5 y + cosh y, Next we use the identity fx = F c F x, Eq. and Proposition. to obtain ξs, χ = M which ends the proof. Corollary.. For all t C c F s = e t Γ F Ξt, χ e t = e i y F + i t, ; iy F ydy, a n P n t where the generating function for coefficients {a n } is a n n! xn = sin x + cosh x. 3 Proof: A rigorous proof and the integral representation for a n can be obtained by expanding the confluent hypergeometric function in in Meixner-Pollaczek polynomials see ]: e iy F λ + it, λ; iy = n= n n λ n P λ n t y n. However we decided to present here a more intuitive argument, which shows why the generating function for the coefficients a n is necessarily the same as the function in the integral representation up to a simple change of variables. 6
First we assume that function Ξt, χ e t lies in the Hilbert space B and we expand it in the orthogonal basis given by Meixner-Pollaczek polynomials: Ξt, χ e t = n a n P t n. Using the orthogonality relation for the Meixner-Pollaczek polynomials see ] we find that the coefficients a n are given by a n = n n! Γ + n Ξt, χ e t P t n Γ + i t dt. R The generating function for {a n } is computed using Eq. : a n e ix n! xn = F + i t, ; ix Ξt, χ e t Γ + i t dt, 3 R and using Proposition. and the integral representation we find that the above integral must be equal to sinx+ cosh x. Corollary.3. For all t C Ξ t, χ e t bn i n = n! + i t + b ] n i n n n! i t, 5 n where the generating functions for coefficients {b n } is b n n! xn = e i +i sin x + x cosh x. Proof: Again we start with the integral representation and rewrite it as Ξ t, χ e t = Ψt + Ψt, where Ψt = e 3i e iy F + it, ; iy cosh dy. y Next we use the definition of the confluent hypergeometric function and expand it in the power series in y see ]. Integrating term by term we find that the coefficients b n have the following integral representation: b n = e 3i n n e iy y n cosh dy. 6 y One can find using the above formula that for n large b n n α e n for some α, thus the series 5 converges for all complex t. The exponential generating function for the coefficients {b n } is computed using Eq. 7. 7
5 Riemann zeta function We adopt the following standard definitions see ],5]: ξs = ss s Γ s ζs and Ξt = ξ + it. Our main result in this section is the following integral representation for Ξt: Theorem 5.. For all t C Ξte t = cos t sin + + t y sin + ye i y 3 F + i t, 3; iy dy. 7 e y + Proof: Define function fx = e x x. The first step is to find that M 3 fs = ξs, Res,. ss Again, the functional equation ξs = ξ s follows from the fact that f is invariant under Fourier sine transform F s see ] and Eq.. Using Eq. 9 we find the fractional sine transform F s where φy = e i y + i have see ] φy F y = F s fy = y sin + e y + + i of function fx: φy φy, 9 Φ e i y. Note that function φy is analytic, and as y + we e i y i y + O M 3 F s φs = i e i, thus φy is in the Hilbert space A. Next we find that y s s, M 3 s F s φs = i e i s, and to finish the proof we only need to combine Eqs., 9, and Proposition.. It is interesting to note that Eq. 7 is essentially equivalent to the integral representation Ξt = + t Υ + it + Υ + it ], where Υs is defined by Υs = e i s s s Γ s L e ix sinh x x s dx. and the integral is taken along the line L = e i R + i see 5]. One can obtain formula 7 by applying Plancherel theorem for sine transform to the functions inside the integral in Eq..
It is also of interest to compare integral representation 7 with the well-known Riemann formula see 5]: ξs = + ss x s s + x e nx dx = x n s = + ss Γ s, n + Γ, n ] n s. n s n The following proposition shows that Eqs. and 7 are just extreme cases α = and α = correspondingly of the more general result: Proposition 5.. For all α, ] ξs = se i α s + se i αs + ss Γ s, e i α n + Γ s, ei α n ] n s 3 n s n and Ξte α t = cos α t sin α + + t where ϑy = Re e 3i + αi i y exp in e iα nye iα n ye i y F 3 + i t, 3 ; iy ϑydy, Proof: To derive 3 one should start with the function ψy = n e iny = θ 3, y and follow the lines of Riemann s proof see 5], but take Mellin transform along the line y e iα R +, α, ]. Formula is obtained from 3 with the help of expression for the incomplete Gamma function as the Laplace transform of the confluent hypergeometric function see ]: Γ s, e i α n n s = ie i α s ]. ye i y 3 F + i t, 3; iy exp in e iα nye iα dy. Next we derive an expansion of the Riemann Xi function in the Meixner-Pollaczek polynomials see 9] for the detailed analysis of the coefficients of this expansion and zeros of its partial sums. Corollary 5.3. For all t C Ξte t = cos t sin where the generating function for coefficients {a n } is a n n! xn = x sin x + tanh x + Φ i + + t a n P 3 n e i i x ix+ e Φ t, 5 e i ] i x ix e. 9
Proof: Again we start with Eq. 7, expand the confluent hypergeometric function in the series of Meixner-Pollaczek polynomials see Eq. and integrate term by term to find that the coefficients are given by y sin + a n = n n +!! e y + yn+ dy. The exponential generating function for {a n } is computed using Eq.. Following the lines of the proof of Corollary.3 we obtain the following expansion of Ξte t in rising factorials: Corollary 5.. For all t C Ξte t = cos t sin + + t bn i n 3 n! + i t + b ] n i n 3 n n! i t, n where the generating function for the coefficients {b n } is b n Acknowledgment: 3i n! xn = e 6 e ix Φ x e i x + eix cosh ] x sinh. x The first version of the manuscript was completed while the author was a Postdoctoral Fellow at the Department of Mathematics and Statistics, McMaster University. The author would like to thank an anonymous referee for many helpful comments. References ] H. Buchholz. The confluent hypergeometric function. Springer-Verlag, 969. ] D. Day and L. Romero Roots of polynomials expressed in terms of orthogonal polynomials. SIAM J. Numer. Anal. 3, 5 5, 969-97. 3] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi. Higher Transcendental Functions, Vol.,, McGraw-Hill, New York, 953. ] I.S. Gradshteyn, and I.M. Ryzhik. Tables of integrals,series and products. 6th edition, Academic Press. 5] A. Iserles, and S.P. Norsett. Zeros of transformed polynomials. SIAM J. Math. Anal. Vol., No. 99, 3-59.
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