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# Tag Archives: Riemann zeta function

## 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

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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
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## The Riemann and Hurwitz zeta functions and the Mellin transform interpolation of the Bernoulli polynomials

This entry (expanding on the Bernoulli Appells entry) illustrates interpolation with the Mellin transform of the Bernoulli polynomials and their umbral inverses, the reciprocal polynomials, giving essentially the Hurwitz zeta function and the finite difference of , both of which … 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
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## Bernoulli, Blissard, and Lie meet Stirling and the simplices: State number operators and normal ordering

A set of identities that encapsulates relations among the Bernoulli numbers, the Stirling numbers of the first and second kinds, and operators related to the umbral calculus of Blissard and his contemporaries: Decoding:

Posted in Math
Tagged Appell sequences, Bernoulli polynomials, Compositional inverse, Conjugation and derivation, Differential operators, Finite difference operator, formal group laws, Hurwitz zeta function, Lie derivatives, Lowering operators, Multiplicative inverse, Raising operators, Riemann zeta function, Sheffer sequences, Stirling numbers, Umbral calculus

## Riemann’s Jump Function J(x) for the Primes

Dirac’s Delta Function and Riemann’s Jump Function J(x) for the Primes presents Riemann’s jump function for counting the primes as introduced in H. M. Edward’s Riemann’s Zeta Function (Dover, 2001), couched in terms of the Dirac delta function and the inverse Mellin transform.