### Recent Comments

### Categories

### Meta

# Tag Archives: Mellin transform

## The Riemann Zeta and the Calculus

(Under construction: Reprising investigations over several years.) By virtue of the relation between the values of the Riemann zeta function at the negative integers, , and the Bernoulli numbers and between the Bernoulli polynomials and the partial sums of the … Continue reading

Posted in Math
Tagged annihilation operator, Bernoulli numbers, Bernoulli polynomials, Beta integral, complex Cauchy contour integral, Confluent hypergeometric functions, convolution integral, creation operators, Differential operators, Digamma function, Dirac delta function, Fractional calculus, gamma function, Generalized Laguerre functions, generalized Laguerre polynomials, harmonic numbers, infinigen, infinitesimal generator, lowering operator, Mellin convolution, Mellin transform, operational calculus, Raising operators, Riemann zeta function, Sheffer Appell polynomials
Leave a comment

## Fractional Calculus, Gamma Classes, the Riemann Zeta Function, and an Appell Pair of Sequences

The background info and comments for the MSE question Lie group heuristics for a raising operator for and the MO question Riemann zeta function at positive integers and an Appell sequence of poylnomials introduce an Appell sequence of polynomials containing … Continue reading

Posted in Math
Tagged Appell sequences, Associated Laguerre polynomials, Bell polynomials, Confluent hypergeometric functions, Convolution operators, Creation and annihilation operators, Cycle index polynomials, Differential operators, Digamma function, Falling factorials, Fractional calculus, Gamma classes, Gamma genus, Infinitesimal generators, Inverse Mellin transform, Mellin transform, Psi function, Raising and lowering operators, Riemann zeta function, Rising factorials, Umbral calculus, Umbral compositional inverse pair
Leave a comment

## Fractional calculus and interpolation of generalized binomial coefficients

Draft Interpolation of the generalized binomial coefficients underlie the representation of a particular class of fractional differintegro operators by convolution integrals and Cauchy-like complex contour integrals.

## Newton Interpolation and the Derivative in Finite Differences

Relations between the normalized Mellin transform (MT) and Newton interpolation (NI) can shed some light on the validity of a finite difference formula for the derivative alluded to in the MathOverflow question MO-Q: Derivative in terms of finite differences. From … Continue reading

## Bernoulli Appells

The defining characteristic of the Bernoulli numbers operationally is that they are the basis of the unique Appell sequence, the Bernoulli polynomials, that “translate” simply under the generalized binomial transform (Appell property) and satisfy (for an analytic function, such as … Continue reading

Posted in Math
Tagged Appell sequences, Cycle index polynomials, Differential operators, Faulhaber's formula, formal group laws, Hurwitz zeta function, Lagrange inversion, Lah polynomials, Mellin transform, Pincherle derivative, Raising operators, Riemann zeta function, Sheffer sequences, Simplices, Stirling numbers, Symmetric polynomials, umbral compositional inverse
4 Comments

## Differential Ops, Special Polynomials, Binomial Transforms, Interpolation, and the Inverse Mellin Transform

Differential Ops, Special Polynomials, Binomial Transforms, Interpolation, and the Inverse Mellin Transform: (pdf, under construction. ) Relations between differential operators represented in the two basis sets and and the underlying binomial transforms of the associated coefficients of the pairs of … Continue reading

Posted in Math
Tagged Associated Laguerre polynomials, Bell polynomials, Binomial transform, Differential operators, Euler transformations, Falling factorials, Finite differences, Generalized Dobinski relation, Inverse Mellin transform, Mellin transform, Modular forms and eigenfunctions, Newton interpolation, Number operator, Permutahedra face polynomials, Stirling numbers of the second kind, Touchard polynomials, Umbral composition
1 Comment