Appells and Roses: Newton, Leibniz, Euler, Riemann, and

Symmetric PolynomialsTom Copeland, Cheviot Hills, Los Angeles, Ca. Oct. 8, 2020

Classic relations among reps of a polynomial in terms of its zeros or a meromorphic function in terms of its zeros and poles, the symmetric polynomials/functions (the elementary, complete homogeneous, and power sum), matrix determinants, the Newton (Waring-Girard-Faber) identities, a couple of composition polynomial sequences (the Stirling partition polynomials of the first kind, a.k.a. the cycle index polynomials--CIPs-- of the symmetric groups, and the Faber partition polynomials), and the formalism of Appell polynomial sequences are presented here and tested on the Riemann function as well as a combinatorial geometric interpretation of the CIPs based on walks on complete graphs, a.k.a. mystic roses.

The characteristic polynomial of degree of a square matrix of rank is defined through the determinant

where are both the eigenvalues of and the roots (zeros) of the polynomial.

Expanding this gives the polynomial in terms of the elementary symmetric polynomials as

(Examples of the elementary symmetric polynomials for three indeterminates are given below.)

Note that

and

This polynomial can readily be re-expressed in terms of the power sum symmetric polynomials as

Now turn to the Taylor series expansion

where are the Stirling partition polynomials of the first kind (cf. OEIS A036039, the Lang link has a compilation) also known as the cycle index polynomials of the symmetric groups . These partition polynomials naturally pop up in discussions of Chern, Pontryagin, and other characteristic classes whose invariance under various transformations reflect the invariance of the determinant of a matrix under conjugation by an inverse pair of matrices. Confer MO-Qs “Canonical reference for Chern characteristic classes,” “Geometric / physical / probabilistic interpretations of Riemann zeta( )?”, “Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants,“ and

“Understanding a quip from Gian-Carlo Rota” as well as other refs in the OEIS entries listed in this set of notes..

To make associations with Taylor series and Appell polynomial sequences, another useful rep of the normalized polynomial in terms of a truncated e.g.f. is

with and for .

Inspecting the identities

we can identify

the Newton identity giving the elementary symmetric polynomials in terms of the power sum polynomials (traces) massaged by the Stirling partition polynomials of the first kind.

Logarithmic differentiation of the reps gives

We can extract the power sums from the Stirling partition polynomials using the Faber polynomials of OEIS A263916, given by the generating function

Therefore,

so

giving the power sums in terms of the elementary symmetric polynomials massaged by the Faber polynomials.

The associated complete homogeneous symmetric polynomials are defined by the generating functions

giving

and, therefore, the intertwined recursion relations via the convolution identity

Logarithmic differentiation gives

We can identify from the generating functions above that

and

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Example identities for the symmetric polynomials / functions:

1) Elementary in terms of power sums via Stirling

2) Complete homogeneous in terms of power sums via Stirling

3) Power sums in terms of elementary via Faber

4) Power sums in terms of complete homogeneous via Faber

5) Complete homogeneous in terms of elementary via the reciprocal partition polynomials (OEIS A263633)

Switch and for the reverse relations.

6) Some elementary symmetric polynomials in three indeterminates

7) Some complete homogeneous symmetric polynomials in three indeterminates

______________________________________________ For characteristic polynomials with real zeros, if we use complex rather than real as our independent variable, we can tease out the roots of the polynomials as a plot of delta functions with a suitable Cauchy integral and Fourier transform as

and

Integrating gives a staircase function with steps locating the roots with a height proportional to their multiplicity. A windowed Fourier transform with limits would give sinc functions rather than delta functions. Using another function other than the complex exponential, such as a narrow gaussian, would give a sum of translations of the function, each centered at a zero. See my post “Jumpin’ Riemann” for an illustration._____________________________________________

Let's connect these generating functions to the Appell formalism and thence to a simple raising operator for generating these important partition polynomials.

First, a partial review of Appell polynomial Sheffer sequences.

A generic Appell sequence has an e.g.f. of the form

where

with and

The lowering operator for any Appell sequence is ; that is,

Binomial convolution with a regular variable translates, or shifts, the Appell polynomials as

and binomial convolution with an umbral variable performs a generalized shift as

These identities follow from multiplication of the respective e.g.f.s

The Appell polynomials can also be generated operationally by

Now consider the umbral dual Appell sequence defined by the reciprocal of

This reciprocal relationship imposes several equivalent identities between the dual pair of Appell sequences. They are an umbral inverse pair; i.e.,

In particular, for ,

This can be shown several ways. One is operationally by

Likewise,

For more on the algebra enveloping these umbral inverse pairs, see my post "Appell Matrices."

The raising / creation operator for an Appell sequence such that

is given by

where .

We can derive the raising op formula through the Graves-Pincherle commutator derivative. The basic Pincherle derivative follows from the following action;

therefore,

so for an analytic function

Then

Then the raising op for can be expressed several ways as

Neatly, the raising op is a conjugation of the prototypical raising op of the fundamental Appell

sequence, the powers . If we look at the Laplace transform and its inverse, multiplication by becomes the differentiation , differentiation by becomes multiplication by , and the conjugation in space looks somewhat like a gauge transformation in space of (see my post "Dirac Appell Sequences'').

Now construct the Appell sequence with the moments

so that

Then the raising op is

generating the Appell sequence with moments

Check that repeated action of this raising op generates the first few listed above.

The Appell sequence may also be expressed as

but convergence of the traces is only guaranteed for polynomial .

The umbral compositional inverse sequence has the e.g.f.

with raising op

Note the simple difference in sign of the differential part of the raising ops for the Appell sequence and its umbral inverse (reciprocal) Appell sequence. Also note

has the form of an Appell e.g.f. but with replaced by , i.e.,

with

and raising op

Restoring ,

with the raising op

and as an Appell sequence in the shift property applies,

Changing signs our particular Appell sequence becomes

The umbral inverse sequence to as an Appell sequence in has the raising op

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For a quick sanity check:

and with a change in signs

therefore, umbral composition gives

The lowering op is for both sequences, as quickly verified.

All peachy keen.______________________________________

Now apply to the cubic generating function with the roots/zeros :

The moment e.g.f. is

The series expansion

diverges for , where the argument of a logarithm becomes negative, i.e., a branch cut is encountered, in this case for , and converges otherwise.

The Appell polynomial sequence e.g.f. is

and the raising op is

Let's verify that we can regenerate the Taylor series for this instance of a moment e.g.f. using the raising op.

Rather than do the algebra for this last coefficient/moment, let's spot check using the values several ways.

and

all in agreement.

Let's check that the next coefficient in terms of the power sums vanishes:

All jim-dandy.

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The specific moment e.g.f. actually can be viewed as a generic e.g.f. with

coefficients being equated to a generic o.g.f. with the generalized coefficients

through , and these generalized need not be related to poles or zeros of a function--just the generic relation between an e.g.f. And an o.g.f. with signed coefficients. The generalized power sums are then determined though the logarithmic derivative to be the coefficients of the raising op of the Appell sequence and need not be identified as traces as above.

Now let's see how a general recursion formula for the Appell sequence is encoded in the action of the raising op

where . Then

which evaluates at the origin to

or

giving a recursion relation for either the -th elementary symmetric polynomial/function in terms of the lower order ones and the power sums, or the -th power sum in terms of lower order ones and the elementary symmetric polynomials/functions

The first few orders are

and conversely

The recursion relations also follow from

The same machinations apply to the umbral inverse Appell sequence

with the raising op

where . Then

which evaluates at the origin to

or

giving a recursion relation for either the -th complete homogeneous symmetric polynomial/function in terms of the lower order ones and the power sums, or the -th power sum in terms of lower order ones and the homogeneous symmetric polynomials/functions

The first few orders are

and, conversely,

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Now apply this Appell apparatus to the reciprocal of the Euler gamma function by

choosing and . Then we have the the raising op

with the digamma function. In this case has simple poles at the negative natural numbers and simple zeros there. Neither function can be written as a

simple product formula as for polynomials since is divergent, but there is a Weierstrass factorization. We do not need this since we can immediately identify from the raising op the Euler-Mascheroni constant. The other are convergent traces related to the simple poles or simple zeros; specifically,

for . Then

for

Note

so

Letting over the Riemann surface for the natural logarithm, then

and

implying

is an infinitesimal generator for a fractional derivative acting on functions analytic on a Riemann surface.

The lowering operator for the Appell polynomial sequence becomes the

lowering operator for the function sequence under the transformation . The general Graves-Pincherle commutator derivatives for lowering and raising operators of sequences for which are

and

Consistently,

Furthermore, since and are the raising and lowering ops for the power series ,

so

Let's corroborate this through the digamma rep

Then

(For more info and other proofs framed in terms of rather than , which differ by a sign in the differential component, see the MSE-Q "Lie group heuristics for a raising operator for

".)

Formally (i.e., ignoring convergence issues), this applies to generic raising ops to define generic divided power shift / translation ops via

with

so

giving an infinitesimal generator for a generic . Since the Appell raising / creation op can always be associated with that for the Stirling partition polynomials of the first kind, we have connections among the combinatorics of cycle index polynomials of the symmetric groups, infinitesimal generators / Lie vectors, and generalized divided power shift operators.

(For more info, see my post "The creation / raising operators for Appell sequences" .)

Note: $t!/p^{t+1} = (1/p) t! (1/p)^t = x t! x^t$ for $x = 1/p$ related to Laplace transform.

or $(t+1)!/ p^t = (t+1)! x^t$ for $x = 1/p$.

Now apply this apparatus to the entire Landau-Riemann xi function.

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Consider an entire complex function whose denumerable complex zeros lie on the vertical critical line at a minimum height of and are reflected through the real axis. Let's further assume the function is represented by the power series and product formula

Let's look at the Appell series

Determining the coefficients for the raising op via the log derivative,

for , the height of the zero closest to the real axis, and

for , but by the symmetry of the zeros, the odd power sums vanish, giving

As in the arguments above, we can use this op to generate the coefficients and readily show that the odd indexed coefficients vanish. Recall that and

Our raising op has only odd powers of the derivative and no constant term, so it will change each polynomial that is even into an odd polynomial and vice versa. We begin with an even polynomial and obtain an odd polynomial and this alternation in symmetry continues as well as a change of sign of the constant term in successive even polynomials, so for odd. Consequently,

is a real-valued, even entire function for real if is convergent, where we sum now only over the complex zeros above the real axis with the imaginary parts of these zeros and, therefore, the real zeros of . To obtain , we assumed but then

turned into the op and our function into a differential op with no concomitant convergence issue.

From the Appell formalism, we know

where for sums over the positive zeros and vanishes for odd , are the Stirling partition polynomials of the first kind, or CIPs, discussed above. Since the odd power sums vanish, the coefficients may also be expressed in terms of the Pontryagin polynomials

presented in OEIS A231846:

where are the odd double factorials of OEIS A001147.

Conversely,

where are the Faber polynomials, with the first few given above.

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Let’s test this apparatus with the Riemann xi function and demonstrate the efficacy of estimating the coefficients of its Taylor series from the power sums of the reciprocals of its zeros and vice versa.

The Landau-Riemann xi function investigated in "Relations and positivity results for the derivatives of the Riemann ξ function" by Coffey can be used to define the real-valued, entire, even (recall ) function

which can be expressed by a simple Hadamard product (a Weierstrass factorization/product) formula of the form above and expanded in the Taylor series

Numerical estimates of the first few derivatives of are given in Coffey (as well as some different ways to compute them).

Titchmarsh in his classic book On the Theory of the Riemann Zeta Function has, on p. 18, Eqn 2.1.14

and on p. 30 he states that it is an even integral function of order 1, whose exponent of convergence is 1. "Hence has an infinity of zeros, whose exponent of convergence is 1. The same is true of " In his Theory of Functions on p. 249 is

Theorem 8.22: If are the moduli of the roots of , then the series is convergent if

in an earlier paragraph is called the order of the integral function .

The absolute contribution of a zero of , , and its complex conjugate to the sum of the

inverse squares of the zeros is with for the real zeros. This is less than , so the trace of the paired inverse squares, even including any complex zeros if they were to be found, is absolutely convergent. Since any potential complex zeros must lie in the critical strip somewhere beyond our current computational skills, they contribute negligibly to any sensible number of significant digits to the Taylor series coefficients. (A meromorphic continuation of the power sums is given by Juan Arias de Reyna in "Computation of the secondary zeta function.")

Gottfried Helms was generous in responding to my MSE-Q "Sums of reciprocals of powers of the imaginary part of the nontrivial zeros of the Riemann zeta function" asking for estimates of partial sums of the power sums of the reciprocal imaginary parts of the nontrivial zeros of the

Riemann zeta function above the real axis for , and in response to the subsequent adjunct MO_Q "Derivatives of Riemann and traces of zeros", Juan Arias de Reyna provided estimates of the traces in good agreement with Helms'.

The estimates (truncated) are

(Helms)

(Reyna)

(Helms)

(Reyna)

For the Riemann xi Appell sequence defined (per discussions above) by the e.g.f.

with , the raising op is

, producing

(use as a quick sanity check),

so evaluating at the origin gives

Comparing Coffey’s estimates for the Landau-Riemann xi function for the derivatives withHelms' and Juan’s estimates:

(Coffey)

(Helms)

(Juan)

and

(Coffey)

(Helms)

(Juan).

Using Coffey’s derivative estimate for and Helms’ or Juan’s for the more rapidly convergent gives

Decent agreement.

Conversely, using the Faber polynomials, we can tease out the traces (as discussed above) from derivatives of

giving from the derivative estimates

in fair agreement with the trace estimates

Now let’s return to the general formalism and elucidate some geometric combinatorial interpretations of the Stirling partition polynomials of the first kind.

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From the Wikipedia article on complete graphs:

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, appeared already in the 13-th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose.

I’ll use paths on the complete graphs to characterize the Stirling partition polynomials of the first kind or cycle index polynomials of the symmetric groups. (Another geometric characterization is provided through trees presented in my notes “Lagrange à la Lah”, but that doesn’t make a connection to determinants.)

The following is basically an elaboration on the ideas sketched in my MO-Q “Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants.”

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety]; the Artin-Mazur zeta function; and a special Reulle (aka dynamical systems or Smale) zeta function, the Ihara zeta function for a graph --all can be expressed in the same basic form:

For graph zeta functions typically is the number of closed walks of steps (with some qualifications) on the graph with vertices and is related to the trace of the power of an edge adjacency matrix. For a vertex adjacency matrix , also (e.g., A054878 and A092297).

You can use the general heuristic to obtain

and then

so

where for .

This last expression is the umbral form for the exponential generating function for the Stirling partition polynomials of the first kind, otherwise known as the cycle index polynomials for the symmetric group (mod signs), presented in the main text.

One example is the Appell sequence in the MO-Q “Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus”

where and for . The polynomials here are not power sums but rather correspond to the Stirling

polynomials with and for .

Let’s explore a geometric combinatorial interpretation of these cycle index polynomials of the symmetric groups via cycles in permutations since a determinant can be defined through the Leibniz formula as signed permutations of its elements.

Reiterating for ease of reference,

Each polynomial can be related to the characteristic polynomial of a matrix with a null main diagonal. In fact, replacing the by gives the characteristic polynomials (mod signs) of the

adjacency matrix of the complete n-graph (see A055137). Note that, for , the term vanishes as it does for the characteristic polynomial of any matrix whose main diagonal is all zeros or whose trace is zero.

For example, for such a 3x3 matrix with the elements with ,the characteristic polynomial is

This is the Leibnitz formula for a determinant in terms of signed permutations of the matrix elements with the elements of the main diagonal equated to , i.e., , and we wish to relate this to

To tease out one geometric combinatorial relationship--mystic roses--begin by picturing a triangle with the vertices labelled 1 to 3. Make closed walks/orbits/loops/cycles/paths traversing the triangle from through and and then to . Assign the 'moment/transition amplitude’ of to each of the two paths of three steps and length three denoted by

, and

of opposite circulation.

Likewise, assign the amplitude to the three paths of two steps and length one combined with one null path,

,

,

;

and an amplitude of to the one combination of the three null paths

.

This generates as an enumeration of combinations of closed paths on the complete graph labelled by the power sums with the number of null paths in the combination correlated to the powers . Paths with opposite circulation, i.e., traversed in the opposite direction, are considered distinct, all vertices must be included in each combination of

paths, and paths in a combination may not intersect each other nor vertices in a single path encountered more than once except for the beginning and ending vertex of the walk, i.e., no loops on loops. This geometrically accounts for the null trace, i.e., the lack of an term, since there is no way of forming the combination of paths or or without leaving out one vertex of the triangle.

Similarly, for

consider a square polygon or tetrahedron with labeled vertices and edges between all pairs of vertices, the complete graph . With cycles/orbits/closed paths of opposing circulation considered distinct cycles, the associated 4x4 determinant generates

1) the six paths of four steps and length four corresponding to

(12,23,34,41) and opposing circulation (14,43,32,21),

(12,24,43,31) and dual (13,34,42,21),

(14,42,23,31) and (13,32,24,41);

2) the three dual paths, each path of two steps and length one for ,

(13,31)(24,42),

(14,41)(23,32),

(12,24)(43,34);

3) the eight paths of three steps and length three combined with one null path for ,

(12,23,31)(44) and (13,32,21)(44),

(12,24,41)(33) and (14,42,21)(33),

(13,34,41)(22) and (14,43,31)(22),

(23,34,42)(11) snd (24,43,32)(11);

4) the six triple paths, each one with one two-step path of length one, and two zero-

step paths of length zero, for ,

(12,21)(33)(44)

(14,41)(22)(33)

(13,31)(22)(44)

(23,32)(11)(44)

(24,42)(11)(33)

(34,43)(11)(22);

5) no possible paths correlated to combinations of three null paths alone such as (11)(22)(33) or (11)(22)(44) since one vertex is left out of the picture, so no term,

6) and finally the one combination of four null paths for ,

(11)(22)(33)(44).

These cycle mappings are depicted by Mark Dominus in his blog post.on “Cycle classes of permutations.”

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Related stuff:

“Polynomial Identities and the Cayley-Hamilton Theorem” by Edward Formanek in which the author states in some sense that “... all polynomial identities satisfied by n x n matrices are consequences of the Cayley-Hamilton Theorem.”

“Cycle indices from the exponential formula and subset / multiset sums divisible by a parameter” by Marko R. Riedel

“On a diagrammatic proof of the Cayley-Hamilton theorem” by Elisha Peterson

“Polya’s enumeration theorem and the symbolic method” by Marko R. Riedel