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# Tag Archives: Fractional calculus

## A Diorama of the Digamma

(Under construction) This series is divergent, so we may be able to do something with it. — Heaviside The divergent series for the pole of the Riemann zeta function is Lets’s use Mellin transform interpolation (essentially the master’s (Ramanujan) master … Continue reading

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Tagged Digamma function, Fractional calculus, gamma function, generalized harmonic numbers, harmonic numbers, infinigen, infinitesimal generator, Inverse Mellin transform, logarithmic derivative, Mellin integral transform, Mellin interpolation, Newton interpolation, power sum polynomials, reciprocal integers, Riemann zeta function, Rising factorial, Taylor series
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## 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
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## 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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## 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.

## Goin’ with the Flow: Logarithm of the Derivative

Goin’ with the Flow: Logarithm of the Derivative Operator is a pdf set of notes under construction on the relations between the commutator of the logarithm of the derivative operator, the Pincherle derivative, Lie operator derivatives, and the two umbrally inverse … Continue reading

Posted in Math
Tagged Bell polynomials, Binomial Sheffer sequences, Commutatator, Confluent hypergeometric functions, Differential operators, Factorial polynomials, Falling factorial, Flow equations, Fractional calculus, Generalized Laguerre functions, Inversion, Laguerre polynomials, Lie derivatives, Logarithm of the derivative operator, Matrix representations, Operator calculus, Pincherle derivative, Rising factorial, Sheffer sequences, Simplices, Stirling numbers, Vandermonde matrix
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## Fractional Calculus, Interpolation, and Traveling Waves

Fractional Calculus, Interpolation, and Traveling Waves: A note (pdf file) on the fractional integro-derivative (FID) regarded as an interpolation of the integral derivatives of the function acted upon–to be more precise, a sinc function interpolation, as in the Whittaker-Shannon interpolation … Continue reading

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Tagged Fractional calculus, Interpolation of operators, Operator calculus
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## A Generalized Dobinski Relation and the Confluent Hypergeometric Fcts.

The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions presents a generalized Dobinski relation umbrally incorporating the Bell / Touchard / Exponential polynomials that is defined operationally through the action of the operator f(x d/dx) … Continue reading