Differintegral Ops and the Bernoulli and Reciprocal Polynomials

* Related stuff* (in addition to the compilation in the pdf):

“Notes on the theorem of Baker-Campbell-Hausdorff-Dynkin” by Michael Muger

“The Campbell–Baker–Hausdorff–Dynkin formula and solutions of differential equations” by Strichartz

“A note on the Baker–Campbell–Hausdorff series in terms of right-nested commutators” by Arnal, Casas, and Chiralt

“Interaction between Lie theory and algebraic geometry” by Shilin Yu

“Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra” by Matthias Beck and Sinai Robins

]]>Juggling Zeros in the Matrix (Example II)

]]>

Example matrices presented in the following pdf are related to Appell polynomials initially explored by Jensen and, subsequently, Polya in their explorations of the real zeros of (more accurately, the hyperbolicity of) Fourier transforms related to the Landau-Riemann xi function, and to the Appell Hermite polynomials and associated Laguerre polynomials found in quantum mechanics and Heisenberg algebras.

]]>The following pdf looks at one common way to define the Faber partition polynomials and an associated Appell polynomial sequence;

*Related Stuff:*

“Representation of Functions by Matrices. Application to Faber Polynomials” by Eri Jabotinsky

]]>Mellin convolution for generalized Hadamard product of functions/power series

]]>(Originally published in Sept. 2019. Inadvertanly deleted in April)

In response to observations initiated by Matt McIrvin of a sum of exponentials of the imaginary part of the non-trivial zeroes of the Riemann zeta function, assuming the Riemann hypothesis is true, as presented on a stream through Mathstackexchange (MSE), Mathoverflow (MO), and the n-Category Cafe. One thread is the MO-Q Quasicrystals and the Riemann Hypothesis posed by John Baez.

The main actors are the Riemann zeta function , the Landau Xi function (aka, the Riemann Xi function with the two poles removed), the von Mangoldt function , the Chebyshev function (aka, the von Mangoldt summatory function), and the Riemann jump function (aka, the Riemann prime number counting function) with Mellin, Heaviside, and Dirac directing, with a cameo by Fourier.

We formally rederive a relationship beween the zeroes of the Riemann zeta function and the powers of the primes (PP henceforth will be our acronym for powers of the primes or prime powers, the , where is a prime and ) that manifests itself in spikes observed at locations of the imaginary part of the nontrivial zeroes–Dirac delta functions arising from taking a Fourier transform of the derivative of a function that sums over the log of PP, the Mangoldt summatory function, aka, the Chebyshev staircase function, aka, the morphed Riemann jump function for counting the primes.

In fact, the Riemann jump function and the Chebyshev staircase function are two sides of the same coin. The Riemann is a sum of Heaviside step functions, , with the edges of the steps located at the powers of the primes (PP) while the Chebyshev gives the same but with edges located at the log of the PP with different step lifts. One can simply be rewritten into the other with a change of parameters. Consequently, taking the derivative of the functions gives us Dirac delta functions located at PP or, alternatively, their logs. And both are directly related to an inverse Mellin transform of the logarithmic derivative of the Riemann zeta.

**Basic Algorithm: Logging product formulas for polynomials**** **

**Step I:** Factor the polynomial or rational function,

**Step II**: Take the logarithmic derivative,

.

**Step III:** Take the inverse Mellin transform:

for , i.e., putting the line of integration to the left of all the zeros, and closing the contour counter-clockwise to the right,

.

**Step IV: **Substitute for for obtaining,

**Step V: **Extend the result as an even function to ,

**Step VI**: Take the Fourier transform to obtain Dirac delta functions at the zeros and the pole of and their negatives,

The basic algorithm is to take logarithmic derivatives of product formulas for a “polynomial” to get a sum of simple poles at the zeroes , or at some other parameters characterizing the “polynomial”, and then applying the linear inverse Mellin transform to turn these poles into a sum of exponentiated terms . Making a simple change of variable gives us a sum of exponentials of the zeroes , or the other parameters–tantamount to taking an inverse Laplace transform of the simple poles.

Our “polynomials” are the Riemann zeta, expressed by the Euler product formula for zeta in terms of the PP , and its equivalent the Landau Xi function, expressed by the Hadamard product formula in terms of its nontrivial zeros, which are the same as zeta’s. The Landau Xi function is the Riemann zeta function with its pole and trivial zeroes removed by multiplying by some fairly simple factors–the most complicated being a gamma function whose singularities remove the simple zeroes (remove them initially but reintroduce them in the subsequent analysis through the poles of the digamma function–the logarithmic derivative of the gamma function.)

In the final analysis we have a relation among the locations of PP through the derivative of the Riemann jump function; the locations of the log of the powers of the primes through the derivative of the Chebyshev function, both derivations related to the inverse Mellin transform of the logarithmic derivative of Euler’s product formula; and the locations of the non-trivial zeros of the Riemann zeta function through the inverse Mellin transform of the logarithmic derivative of Hadamard’s product formula for the Landau Xi, in terms of the nontrivial zeros.

*Detailed Formal Analysis*

Let’s work through the analysis in more detail.

With the Heaviside step function and designating a prime,

where are the natural numbers, since

for with a natural number and vanishes otherwise. Then employing the Dirac delta function

so

where .

Now look at a property of the Mellin transform. If

,

then for suitably behaved functions

and, from the earlier post on the Riemann jump function,

implying, for ,

Therefore,

Now make the change of variable , giving

which agrees with the rep above of the zeta fct. ratio and the inverse Laplace transform rep of a delta fct.

The last linear contour integral is essentially an inverse Laplace transform. At this point, it would be good to do a sanity check by numerically evaluating the line integral over finite limits by replacing the infinities by , some finite extent. This will give sinc functions for the delta functions. But, let’s continue.

To get some relation to the non-trivial zeros, multiply the top and bottom of the zeta function ratio to convert the denominator to Landau’s , an entire function symmetric about , with no trivial zeros, and the same non–trivial zeros as the Riemann zeta :

.

Let’s see if we can tease out a by taking the derivative of and comparing it to the numerator of our ratio. We can then make use of the Hadamard product representation of this log to relate this to pairs of the non-trivial zeros.

Differentiating term by term and using the digamma or psi function (due to a conflict of notation with the Chebyshev function, I shall use for the digamma function):

so

and our ratio becomes

.

The Hadamard product formula gives

where the sum is over the zeros above the real axis and the lower zeroes are entered through taking the complex conjugate.

Taking the derivative,

Now appealing to Mangoldt’s formula, formally derived in Appendix II from the formulas above, we have for

Taking the derivative and relating this to our other expression above,

Multiplying both sides by , and suppressing the Heaviside step function, gives

and, letting for ,

,

and,

.

We can extend this equation to as an even function on both sides by changing to on both sides and averaging the two equations together. Taking the Fourier transform w.r.t. will give an odd function of Dirac delta functions on the LHS located at absiccas equal to the imaginary part of the nontrivial zeros.

The 2007 pdf “What is the Riemann hypothesis?” by Mazur and Stein contains plots of partial sums of the and a discussion. I assume their book extends and elaborates on these fantastic facets of the Riemann zeta.

**Appendix I: Mellin Transforms an****d Inverse **

Given .

,

and for suitably chosen .

**1) ** for so that the upper evaluation vanishes.

For ,

for if we truncate the vertical integration line with the infinities replaced by some positive finite and close the contour counter-clockwise to the left with a semicircle of radius for , the closed contour contains no singularities, so the contour integral evaluates to and the integral along the semicircle tends to zero as tends to infinity, so the integration over the vertical line evaluates to zero. On the other hand, if we close clockwise to the right with a semicircle, which introduces an overall negative sign, with , the closed contour contains a simple pole at the origin and evaluates to unity while the integral along the semicircle vanishes in the limit as tends to positive infinity.

**2)** for so that the lower eval vanishes.

For ,

.

**3) ** for .

For .

.

**4) **,

for , and for ,

**5 a.) **Now to tackle the inverse Mellin transform of the digamma function using the representation

,

for (or 0 for the integral rep?), where is the Euler-Masheroni constant.

,

and

.

for , but analytically continues to all complex numbers except the negative natural numbers .

Therefore,

**5 b)** ,

so the inverse Mellin transform for closing clockwise to the right for gives

Note the Mellin transform of this expression would have to be regularized to obtain the digamma expression again. This is a common occurence. In fact, regularization is applied to get a Mellin transform for the continuation of the gamma function to the left of its singularities, and the Euler integral expression for the digamma itself is a differently regularized Mellin transform.

**5 c)** For Mangoldt’s explicit formula, we need to evaluate a slightly different inverse transform of a digamma function:

for closing clockwise to the right for gives

**6) **A general Mellin transform relation between the Mellin transform of a function and that of an integral of the function through integration by parts:

so integrating over from to and rearranging terms,

where for suitably chosen .

Alternatively, with ,

with .

**7)** For example, for ,

and

giving

for .

This is consistent with

and

for and ,

and with

.

**8)** To evaluate double poles:

for and suitably decaying at infinity to the right of with no other poles or branch cuts in that region. Depending on , the evaluation could be extended to .

(At first I thought I needed to eval double poles, but I found a way to circumvent it. I leave this for illustrative purposes of the properties of the inverse Mellin transform.)

(We could also collapse our integration line to just beneath and above the real axis, like a Hankel contour, to the right of and generate an integral along the real axis containing the derivative of a Dirac delta function.)

*Appendix II: **M***angoldt’s explicit formala for the Chebyshev function **

To relate the analysis here to other derivations, note that , so

and , and, in particular,

becomes

under the obvious change of variable from to .

In the main body of this post, we find the relation between Chebyshev’s function, or Mangoldt’s summatory function, and an inverse Mellin transform of the log derivative of the Riemann zeta. Evaluating this transform, and equating it to other expressions above we arrive at Mangoldt’s explicit formula.

For ,

Evaluating term by term and using the identity for the digamma in Appendix I Example 5c,

Since symmetry gives , and according to “Relations and positivity results for the derivatives of the Riemann function” by Coffey

,

.

*Appendix III: Dirac delta combs and approximations*

Consider the nascent Dirac delta function given by

.

Then the sifting property holds on functions continuous around the origin and suitably behaved elsewhere:

.

This can be shown for suitable functions by using the Fourier convolution theorem and then taking the limit, but works for a wider class of functions also.

Note that taking the Fourier transform over finite limits gives us our oscillating function, a nascent delta function, rather than a sharp spike.

Now consider a sum of exponentials of uniformly spaced purely real numbers symmetric about the origin when the spacing between the numbers is so that for and . Then

for any integer , so the sum of exponentials gives a periodic function which behaves about as

which tends to

as and tend to infinity with a fixed constant, giving us our Dirac comb

.

Of course, the zeros of the Riemann zeta are not evenly spaced, so we can’t expect to find a Dirac comb in our case, but the analysis does suggest at best summing over a finite number of zeros will give us oscillating sinc functions rather than Dirac delta functions.

**Appendix IV:*** ** Riemann’s explicit formula* (added Oct. 2, 2019)

Riemann derived

where , the logarithmic integral, and, as above, denotes the nontrivial zeros.

From the main text above, we have

,

so for , we have from the analysis in the main text

,

and from differentiating Riemann’s explicit formula, we obtain consistently

.

**Appendix V:** **More basic Mellin transform properties**

Dirichlet series are best thought of as inhabiting the inverse Mellin transform space and as the dual of a Dirac delta distribution in real space as shown in my earlier post on the Riemann jump function:

Then

Note this becomes if or, equivalently, . See the famous example in my answer to the Math Stackexchange question “Does the functional equation have any nontrivial solutions … ?”

What is the inverse Mellin transform of for ?

** Appendix VI**:

Under construction:

1) Sifting property

We define the Dirac delta “function” by its sifting property acting within an integral on a function suitably behaved about a small neighborhood of the point where the argument of the delta function vanishes:

,

so we have

.

The singularity at the origin is no problem if since by integration by parts

,

so

.

This has no singularity at the origin if doesn’t.

Now taking the derivative of the sift equation above, we obtain

, so

.

(In fact, the relevant functions dealt with in the main text are null at the origin and for some interval to the right of it.)

Changing variables we derive other properties.

2) Even symmetry property

Let , then

,

, but also

,

so the Dirac delta is an even “function”; i.e.,

.

3) Reciprocal scaling property

With ,

let . Then for ,

,

and for ,

Therefore, .

4) Composition property

Let with only one zero and that at , then since we are only concerned with where the argument of the delta vanishes

.

.

5) The derivative property

Clearly the derivative of the delta is odd since the delta is even:

, so

.

Use the limit of the Newton-Fermat quotient,

in the limit gives

.

Similarly, by a shift, or change of varisble,

.

6) The BYOYB property

Beware! The magical delta can bite you on your butt at a moment’s notice if you aren’t careful. For example, the nascent Dirac delta function discussed above

is sometimes said to approach the Dirac delta as approaches infinity yet it lacks the important symmetry

that follows from the properties above. This apparent quandary is resolved once the the nascent function is embedded in an integral:

as

as

**Related Stuff:**

Trying to corroborate my analysis, I found consistent results and extensions

1) “Notes on the Riemann hypothesis” by Peraz-Marco

2) The Prime Number Theorem: a proof outline: at the Number Theory and Physics Archives

3) Chebyshev function at Wikipedia

4) Beurling zeta functions, generalized primes, and fractal membranes by Hilberdink and Lapidus

5) Spectral analysis and the Riemann hypothesis by Lachaud

6) Notes on the Riemann hypothesis by Perez-Marco

7) An essay on the Riemann Hypothesis by Alain Connes

Added on Oct. 3, 2019:

8) “The Riemann hypothesis explained” by Veisdal (blog post). A quick general intro to the Riemann zeta, RH, and the prime number theorem.

9) “Riemann’s Explicit Formula” by Sean Li. Derivations with convergence arguments provided.

More

10) “A history of the prime number theorem” by Anita Alexander

11) “A history of the prime number theorem” by Goldstein

12) “Mellin convolution and its extensions, Perron formula, and explicit formulae” by Jose Javier Garcia Moreta. An analysis also based on simple Mellin transform properties.

13) Convergence of Riemann spectrum/Fourier transform of prime powers a MSE question posed by Joe Knapp and answered by reuns

14) “Twenty female mathematicians” by H. Williams. (See the section on Mirzakhani.)

15) “Some remarks on the Riemann zeta distribution” by Allan Gut

16) “A computational history of prime numbers and Riemann zeros” bu Moree, Petrykiewicz, and Sedunova

17) Posts by Markus Shepherd on the Riemann zeta function

]]>for . (Cf. the MathOverflow question Motivation of the Virasoro algebra.)

The function represents a Lie group under composition with respect to the parameter . That is, with roots always chosen as positive and real for in a suitably small neighborhood of the origin,

and the compositional inverse of is simply

The function is also an exponential generating function for the generalized left and right factorials (for positive and negative integral ) presented in OEIS A094638 (mod sign, index shifts, and an additional initial 1 in some cases).

For example,

and the sequence is signed A001147, which I will call the signed right double factorial with an additional initial 1 (cf. A094638). (The numerators and denominators of the reduced fractions are A098597 and A046161, apparently.)

The compositional inverse is

and the sequence is A001147.

We can say the augmented right double factorial as represented in this group is skew invariant under compositional inversion. It is also quasi-invariant under multiplicative inversion with

generating 1,1,-1,3,-15, … .

In contrast, for , we have

generating , signed A007559, the signed, right triple factorial augmented with an initial 1.

The shifted reciprocal gives

generating , the signed left triple factorials A008544 augmented with an initial 1.

Note that the compositional inverse pairs can be related to dual families of trees and also the multiplicative inverse pairs, according to the OEIS entries. See also the MO-Q Combinatorial interpretation of series reversion coefficients and the Gessel link therein, which discusses both multiplicative and compositional inversion. The two types of inversions are also related in general to Hopf algebras/monoids and Koszul duality.

The compositional inverse of can be computed from the coefficients of using the compositional inversion formula A134264 related to free cumulants in free probability theory, non-crossing partitions, and Dyck paths, among other combinatorial constructs. This algorithm allows a transformation of right factorials into left factorials, as does A133437

]]>A formula for computing the structure, or linearization, constants for reducing products of pairs of polynomials of a binomial Sheffer sequence, , is presented in terms of the umbral compositional inverses of the polynomials, . To say the pair are umbral inverses means

which is equivalent to the lower-triangular coefficient matrices being an inverse pair.

This is used to give a simple computation for the first order structure constant of the pair, which determines the associated coefficients of the associated formal group law, defined by

See the earlier post Formal Grouo Laws and Binomial Sheffer Sequences for more discussion on this topic.

Given

the linearization coefficients are defined for by

Equating coefficients of powers of and using standard linear algebra, one finds that a matrix composed of columns containing the coefficients of the umbral compostional inverses of the polynomials multiplying on the left the column vector containing the convolution components of the product gives the linearization components.

The umbral compositional inverse sequence can be generated a number of ways. One is by takiing the matrix inverse of the lower triangular matrix containing the coefficients of the binomial sequence . Another way is by constructing the compositional inverse of either analytically or by using a Lagrange compositional inversion formula and then forming the inverse binomial sequence as

.

A third way is by replacing each coefficients used to construct the original compositional binomial sequence by , the coefficient of the inverse series . Then

.

This is true for the o.g.f. defined as

Construct the associated compositional Lah partition polynomials

These compositional Lah partition polynomials are presented in Lagrange a la Lah Part I and correspond to those of OEIS A130561.

The first few, with our sign designation, are

Take the matrix inverse of the coefficient matrix of these polynomials to obtain the coefficient matrix of the umbral inverse sequence or use A133437 to obtain the umbral inverse series for as with the first few

(Note that these are the refined face polynomials of the celebrated associahedra.)

We can use A145271 to show that all the numerical coefficients are positive integers since has only positive numerical coefficients.

Now use the formula for the Lah compositional partition polynomials with the o.g.f . for , or, equivalently, use

giving

Then construct the upper triangular matrix whose columns are the coeficients of

As a spot check that the two sequences are an umbral inverse pair, note

which is the intended result.

Now evaluate the convolution column vector for the product

so

Then

and the linearization coefficients are

Check:

and

in agreement.

In the earlier post Formal Group Laws and Binomial Sheffer Sequences, I show that the Taylor series coefficient of each is

so let’s check that this gives the same result as above:

which is consistent.

It was shown in the associated MathOverflow Q&A Characterizing Positivity of Formal Group Laws that the coefficients of the FGL constructed from the o.g.f.s above such that has all positive coefficients are all positive. This means all have positive numerical coefficients. Now the question is whether all the linearization coefficients for every product of pairs have only positive numerical coefficients for the given form of the o.g.f. of , as in this example for only.

Adding

and

to the analysis gives

Edit (Jan. 3, 2020):

In fact, Jair Taylor answers this conjecture in the affirmative through Prop. 7.1 on page 54 of his thesis “Formal group laws and colored hypergraphs” and the proof provided in his post of positivity of the associated . The linearization coefficients are the coefficients of the series expansion of .

*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 formula) to interpolate the harmonic numbers , the partial sums of the divergent series, in the hope that we can glean some global numerics of the Riemann zeta. The digamma function and its various avatars will naturally spring forth.

But first these excerpts:

The world of ideas is ‘one’ – i.e., it is a cohesive living organism in which all ‘parts’ interact and where even slight stimulations propagate producing echoes in the (seemingly) distant organism which may be called theories or ‘branches of mathematics’. Similarly, like in the Weierstrass-Riemann principle of ‘analytic continuation,” a change of a (meromorphic) function, even within a very small domain (environment), affects through analytic continuation the whole of Riemann surface, or analytic manifold. Riemann was a master in applying this principle and also the first who noticed and emphasized that a meromorphic function is determined by its ‘singularities’. — Maurin

*Euler was probably the first to see that these series can be applied to number theory. He was in correspondence with C. Goldbach and J.L Lagrange just on number theory questions. His proof of the existence of innitely many primes uses the divergence of the harmonic series* * and using the above fundamental theorem of arithmetic which says that every natural number can uniquely be written as a product of power*s of primes.

**I) ***The digamma function as an interpolation of the harmonic numbers*

The Ramanujan-Mellin-Newton interpolation formula, using the umbral notation lowering of superscripts and Euler’s integral formula for the gamma function, is

.

The shortest route to the Newton series is through the two integrals on the last line where we directly employ an umbral Euler’s integral (UEI)

.

where the notation denotes we must develop a power series rep before umbrally lowering superscripts.

Note for where c is some positive constant (or operator) that this UEI gives as the interpolation of the Taylor series coefficients (TSC) of and that the UEI, as well as the Newton series, gives the th TSC when allowing the interpretation

for which

.

Here denotes the Heaviside step function.

Returning to our Newton series for interpolating , notice that we have not provided a value for yet.

Euler provided the obvious integral formula

,

giving if we perturb about 0.

Taking finite differences of the harmonic integral

Now plugging Euler’s harmonic integral into the Newton series

.

Implying, when we expand , that

,

for .

**II) The digamma meets the Riemann zeta through the log derivative of the rising factorial: Power sum series for the digamma function**

One way of generating such poles is to take a limit as tends to infinity of the logarithmic derivative of a simple product, the rising factorials,

,

and lo and behold we have our digamma or function.

Then in the limit, this becomes

and the Euler-Mascheroni constant materializes. We note that this also shows as .

Such products lie at the foundations of the fomalism of symmetric polynomials/functions–the elementary, homogeneous complete, and power sum symmetric polynomials, and we now use the common maneuver of relating this logarithm of products to the power sums which turn out to be for .

Focusing on the last summand,

.

Then

and we have our anticipated connection of global, yet isolated, numerics of the Riemann function to its behavior just to the right of its single pole.

**III) Summary of formulas for the digamma function**

Reprising, with 20/20 hindsight we have established a connection of the partial sums of the divergent series for , aka the harmonic numbers, through their shifted interpolation, the digamma function, to the values of the Riemann zeta at the natural numbers greater than one. Hidden in the foliage are the rising factorials, whose polynomials contain the Stirling numbers of the first kind as their coefficients, and the power sum symmetric polynomials.

There is an associated sinc function interpolation also :

Collecting our reps of so far, we have

for

for

for

where

,

and

with .

*IV) Mellin transform space and the digamma function*

Now let’s morph our interpolation in the inverse Mellin space of complex numbers back to the real space.

, so

, and

.

There is no convergence problem since

.

Then the inverse Mellin transform gives

,

which is in agreement with the Bateman/Erdelyi Tables of Integral Transforms.

A direct inverse transform gives,

The function summand can not be futher reduced and is to be interpreted as the regularization summand.

Same holds from

Interpreting and checking,

Also

consistent with

.

The equivalent inverse transform (apply inside the line integral below) is

A change of variable transforms this line integral into , so checking consistency by taking the Mellin tranform, for ,

(The digamma manifests in previous entries in connection with the trivial zeros of the Riemann zeta. See the post “Jumpin’ Riemann …”, in which we require another inverse Mellin transform.)

**V) The digamma diff op as a raising/creation op and an infinitesimal generator for fractional calculus**

The digamma has popped up in previous entries as the differential component of a raising op for the Appell polynomials of the gamma genus that form a basis for a fractional calculus and, equivalently, as the differential component of the infinitesimal generator for the fractional (actually, real powers of) integrals and derivatives of that calculus.

*V 1.) Operational calculus, finite differences, and Mellin/Newton interpolation*

is an eigenfunction of the Euler, or state number, op and, by definition, with the eigenvalues and , the falling factorial, respectively; that is,

and

so we can expect both to be intimately connected to the Mellin transform and Newton/Mellin interpolation.

In addition, they are associated to an important pair of umbrally inverse, binomial Sheffer polynomial sequences with significnt combinatorial interpretations–the Bell polynomials and the falling factorials, denoted above, with coefficients the Stirling numbers of the second kind and the first kind, respectively. These can be easily generalized to multinomial partition polynomials–the general Bell polynomials, OEIS A036040, or the refined Stirling partition polynomials of the second kind, and the cycle index partition polynomials for the symmetric groups, or refined Stirlng partition polynomials of the first kind, OEIS A036039.

With the convenience of notation and the fomulaic suggestivity/heuristic benefits provided by the umbral maneuver and with , we have some formulas that lie at the heart of the umbral operational calculus:

so

and, therefore,

We also have our generalized Taylor series from a generalized shift/translation operator acting on :

So, the umbral operational calculus shadows that of the binomial convolution hence the descriptor umbral.

Now back to our Sheffer sequences and their op reps. Define the Bell polynomials operationally through exponentiation of the number op, or Euler op, as

.

Then the e.g.f. for the Bell polynomials is

Note also

since vanishes for any polynomial of order for .

The Bell numbers are given by , so

This is called the Dobinski formula (explained combinatorially by Rota), a particular case of the more general formula

with and .

Operationally, using Taylor series,

so that operating formally on a function

where .

Similarly, acting on a function not necessarily analytic at the origin,

where

Another useful differential formula (a variation on the theme) is

so

where

or, more generally,

Also, when operating on a function analytic at the origin,

The following identity is useful in moving in the reverse direction–obtaining operational identities from Newton series/Mellin interpolations:

so

,

i.e., the falling factorials and the Bell polynomials are an umbral inverse pair, which satisfy

,

with

where

essentially equivalent to multiplication of matrix inverse pairs.

Now we are ready to generalize to evaluations of the action of more complex operators through the inverse Mellin transform, defining

or

where

Then

*V 2.) The digamma in a raising op for an Appell sequence*

This section will incorporate material from several earlier entries and MathOverflow and Math Stackexchange questions I posed:

A.) The Raising / Creation Operators for Appell Sequences, a blog post and pdf.

B.) Riemann Zeta Function at Positive Integers and an Appell Sequence of Polynomials, a MathOverflow question

C.) Lie Group Heuristics for a Raising Operator for , a Math Stackexchange question

D) Multiple Zeta Values Related to Fractional Calculus and an Appell Polynomial Sequence, a MathOverflow question

E) Cycling Through the Zeta Garden: Zeta Functions for Graphs, Cycle Index Polynomials, and Determinants, a MathOverflow question

F) Goin’ with the Flow: The Logarithm of the Derivative Operator, a blog post and pdfmmm

G) Fractional Calculus, Gamma Classes, the Riemann Zeta Function, and an Appell Pair of Sequences, a blog post

H) Fractional Calculus and Interpolation of Generalized Binomial Coefficients, a blog post

I) Fractional Calculus, Interpolation, and Travelling Waves, a blog post and pdf

J) Mellin Interpolation of Differential Ops and Associated Infingens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops, a blog post

K) Superstring Amplitudes, Unitarity, and

Hankel Determinants of Multiple Zeta Values by

Michael B. Green and Congkao Wen