Mellin Transforms

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The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform

R S Dhaliwal - One of the best experts on this subject based on the ideXlab platform.

Mark W. Coffey - One of the best experts on this subject based on the ideXlab platform.

  • bernoulli identities zeta relations determinant expressions Mellin Transforms and representation of the hurwitz numbers
    Journal of Number Theory, 2018
    Co-Authors: Mark W. Coffey
    Abstract:

    Abstract The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, is generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet L series, to Eisenstein series, and to general Mellin Transforms. The Hurwitz numbers H ˜ n occur in the Laurent expansion about the origin of a certain Weierstrass ℘ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general ℘ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence. Other results concern the Matter numbers of the equianharmonic case of the ℘ function.

  • bernoulli identities zeta relations determinant expressions Mellin Transforms and representation of the hurwitz numbers
    arXiv: Number Theory, 2016
    Co-Authors: Mark W. Coffey
    Abstract:

    The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin Transforms. The Hurwitz numbers $\tilde{H}_n$ occur in the Laurent expansion about the origin of a certain Weierstrass $\wp$ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general $\wp$ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence.

  • Mellin Transforms with only critical zeros: Legendre functions
    Journal of Number Theory, 2015
    Co-Authors: Mark W. Coffey, Matthew C. Lettington
    Abstract:

    We consider the Mellin Transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the Transforms have polynomial factors whose zeros lie all on the critical line View the MathML sourceRes=1/2. The polynomials with zeros only on the critical line are identified in terms of certain View the MathML sourceF23(1) hypergeometric functions. These polynomials possess the functional equation pn(s)=(−1)⌊n/2⌋pn(1−s)pn(s)=(−1)⌊n/2⌋pn(1−s). Other hypergeometric representations are presented, as well as certain Mellin Transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory.

  • Mellin Transforms with only critical zeros generalized hermite functions
    arXiv: Complex Variables, 2013
    Co-Authors: Mark W. Coffey
    Abstract:

    We consider the Mellin Transforms of certain generalized Hermite functions based upon certain generalized Hermite polynomials, characterized by a parameter $\mu>-1/2$. We show that the Transforms have polynomial factors whose zeros lie all on the critical line. The polynomials with zeros only on the critical line are identified in terms of certain $_2F_1(2)$ hypergeometric functions, being certain scaled and shifted Meixner-Pollaczek polynomials. Other results of special function theory are presented.

  • Mellin Transforms with only critical zeros: Legendre functions
    arXiv: Mathematical Physics, 2013
    Co-Authors: Mark W. Coffey, Matthew C. Lettington
    Abstract:

    We consider the Mellin Transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the Transforms have polynomial factors whose zeros lie all on the critical line Re $s=1/2$. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. These polynomials possess the functional equation $p_n(s)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s)$. Other hypergeometric representations are presented, as well as certain Mellin Transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory.

B M Singh - One of the best experts on this subject based on the ideXlab platform.

Jon G Rokne - One of the best experts on this subject based on the ideXlab platform.

Johannes Blümlein - One of the best experts on this subject based on the ideXlab platform.

  • structural relations of harmonic sums and Mellin Transforms up to weight w 5
    Computer Physics Communications, 2009
    Co-Authors: Johannes Blümlein
    Abstract:

    Abstract We derive the structural relations between the Mellin Transforms of weighted Nielsen integrals emerging in the calculation of massless or massive single-scale quantities in QED and QCD, such as anomalous dimensions and Wilson coefficients, and other hard scattering cross sections depending on a single scale. The set of all multiple harmonic sums up to weight five cover the sums needed in the calculation of the 3-loop anomalous dimensions. The relations extend the set resulting from the quasi-shuffle product between harmonic sums studied earlier. Unlike the shuffle relations, they depend on the value of the quantities considered. Up to weight w = 5 , 242 nested harmonic sums contribute. In the present physical applications it is sufficient to consider the subset of harmonic sums not containing an index i = − 1 , which consists out of 69 sums. The algebraic relations reduce this set to 30 sums. Due to the structural relations a final reduction of the number of harmonic sums to 15 basic functions is obtained. These functions can be represented in terms of factorial series, supplemented by harmonic sums which are algebraically reducible. Complete analytic representations are given for these 15 meromorphic functions in the complex plane deriving their asymptotic- and recursion relations. A general outline is presented on the way nested harmonic sums and multiple zeta values emerge in higher order calculations of zero- and single scale quantities.

  • analytic continuation of Mellin Transforms up to two loop order
    arXiv: High Energy Physics - Phenomenology, 2000
    Co-Authors: Johannes Blümlein
    Abstract:

    The analytic continuation of the Mellin Transforms to complex values of N for the basic functions $g_i(x)$ of the momentum fraction x emerging in the quantities of massless QED and QCD up to two-loop order, as the unpolarized and polarized splitting functions, coefficient functions, and hard scattering cross sections for space- and time-like momentum transfer are evaluated. These Mellin Transforms provide the analytic continuations of all finite harmonic sums up to the level of the threefold sums of transcendentality four, where the basis-set ${g_i(x)}$ consists of products of {\sc Nielsen}-integrals up to transcendentality four. The computer code {\tt ANCONT} is provided.

  • harmonic sums and Mellin Transforms up to two loop order
    Physical Review D, 1999
    Co-Authors: Johannes Blümlein, Stefan Kurth
    Abstract:

    A systematic study is performed on the finite harmonic sums up to level four. These sums form the general basis for the Mellin Transforms of all individual functions ${f}_{i}(x)$ of the momentum fraction x emerging in the quantities of massless QED and QCD up to two-loop order, as the unpolarized and polarized splitting functions, coefficient functions, and hard scattering cross sections for space and timelike momentum transfer. The finite harmonic sums are calculated explicitly in the linear representation. Algebraic relations connecting these sums are derived to obtain representations based on a reduced set of basic functions. The Mellin Transforms of all the corresponding Nielsen functions are calculated.

  • Harmonic sums and Mellin Transforms
    Nuclear Physics B - Proceedings Supplements, 1999
    Co-Authors: Johannes Blümlein
    Abstract:

    Abstract The finite and infinite harmonic sums form the general basis for the Mellin Transforms of all individual functions fi(x) describing inclusive quantities such as coefficient and splitting functions which emerge in massless field theories. We discuss the mathematical structure of these quantities.

  • On the Mellin transform of the coefficient functions of F(L)(x,Q**2)
    arXiv: High Energy Physics - Phenomenology, 1997
    Co-Authors: Johannes Blümlein, Stefan Kurth
    Abstract:

    The Mellin-Transforms of the next-to-leading order Wilson coefficients of the longitudinal structure function are evaluated.

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