## Witt Differential Generator for Special Jack Symmetric Functions / Polynomials

Exploring some relations among the multinomial coefficients of OEIS A036038 and the compositional inversion formulas of A134264, A248120, and A248927, related to numerous combinatorial structures and areas of analysis, including noncrossing partitions and free probability,  I came across the Jack symmetric functions $J_n^{\alpha}(x_1,x_2, ...)$ in an infinite number of variables as presented in “Some combinatorial properties of the Jack symmetric functions” by Stanley.

Equation a) on page 80 of his paper leads to an umbral generating formula for the related Jack symmetric polynomials (JSP)

$J_n^{\alpha}(x_1,x_2, ...,x_{n+1}) = (\hat{x}_1 + \hat{x}_2 + \cdots + \hat{x}_{n+1})^n$,

where $\hat{x}_k$ is to be regarded as a regular variable until the expression is reduced to monomials at which time it is to be evaluated as $\hat{x}_k^j = s_j (\alpha) \; x_k^j$ with $s_j(\alpha) = 1 (1+\alpha)(1+2 \alpha) \cdots (1+(j-1) \alpha)$, essentially the row polynomials of A094638 comprised of the Stirling numbers of the first kind.  For example,

$J_2^{\alpha}(x_1,x_2,x_3) = (\hat{x}_1 + \hat{x}_2 + \hat{x}_3)^2$

$= \sum_{k=1}^3 \; \hat{x}_k^2 \; + \; 2 \; \sum_{i,j=1 ; i < j}^3 \; \hat{x}_i \; \hat{x}_j = m_{[2]}(\hat{x}_1,\hat{x}_2,\hat{x}_3) \; + \; 2 \; m_{[1,1]}(\hat{x}_1,\hat{x}_2,\hat{x}_3)$

$= s_2(\alpha) \; m_{[2]}(x_1,x_2,x_3) \; + \; 2 \; s_1(\alpha) \; s_1(\alpha) \; m_{[1,1]}(x_1,x_2,x_3)$

$= (1+\alpha) \; m_{[2]}(x_1,x_2,x_3) \; + \; 2 \; m_{[1,1]}(x_1,x_2,x_3)$,

where the polynomial has been expressed in the symmetric monomial polynomials (SMP), easily extended to an indefinite number of variables. The factors multplying the SMPs are the multinomial coefficients of A036038 and remain independent of the number of variables. Each summand of an SMP has the same configuration of exponents and subscripts, allowing the products of $s_k (\alpha)$ to be easily determined and factored out after the umbral evaluation.

Similarly, umbral reduction of $x_k^j = x_j$ transforms $J_n^0(x_1,..,x_{n+1})$ into the partitions of A248120.

MOPS: Multivariate orthogonal polynomials (symbolically)” by Dumitriu, Edelman, and Shuman contain examples for the third and fourth JSPs, but the third has the coefficients erroneously transposed for the factorial.

Using the operator identities in A094638, a Rodriques-like generator for the JSPs can be devised.

$(z^{1+y} \; D_z)^n = z^{ny} \; zD_z \; s_n (y \; zD_z)$,

so

$z^{-1} \; z^{-ny} (z^{1+y} \; D_z)^n \; z = z^{-1} \; zD_z \;s_n(y \; zD_z) \; z = s_n (y)$,

and, with $w_k = 1/z_k^y$

$(\hat{x}_k)^j = z_k^{-1} \; w_k^j \; (z_k^{1+y} \; D_{z_k})^j \; x_k^j \; z_k = s_j(y) \; x_k^j$

$= z_k^{-1} (x_k \; \hat{w}_k \; z_k^{1+y} \; D_{z_k})^j \; z_k$,

with $\hat{w}_k$ treated as independent of $z_k$ with respect to the derivations and, after being passed unscathed to the left of all derivations, is finally evaluated as  $\hat{w_k} = 1/z_k^y$.

Then, with $Q = \prod_{k>0} \; z_k$,

$J_n^y = Q^{-1} \; [\sum_{k>0} \; x_k \; \hat{w}_k \; z_k^{1+y} \; D_{z_k}]^n \; Q$.

This may be generalized just as the Laguerre polynomials are to the associated Laguerre polynomials by conjugating with $Q^u$ rather than $Q$.

A generating function for the full set of JSPs is

$\exp[t\; J.^y] = Q^{-1} \; \prod_{k>0} \; \exp[t \; x_k \; \hat{w_k} \; z_k^{1+y} \; D_{z_k}] \; Q$.

This can be evaluated by noting, with

$\sigma = z^{-y}/(-y)$ and $L_y = z^{1+y} \; \frac{d}{dz} = \frac{d}{d\sigma}$,

that

$e^{\beta \; L_y} f (z) = e^{\beta \; \frac{d}{d\sigma}} \; f[(-y\sigma)^{-1/y}] = f[(-y (\sigma +\beta))^{-1/y}]$.

Then, for $\beta = t \; x_k \; \hat{w_k}$ and $f(z)= z$, the operation reduces to

$[-y(\sigma+\beta)]^{-1/y} = [z^{-y}-t \; y \; x_k \; \hat{w_k}]^{-1/y} = z / [1-t \;y \; x_k \; \hat{w}_k \; z^y]^{1/y}$.

Then the generator gives, since ultimately $\hat{w_k} = 1/z_k^y$,

$e^{t \; J.^y} = \prod_{k>0} \; [1-t \; y \; x_k]^{-1/y}$,

or

$e^{t \; J.^\alpha} = \prod_{k>0} \; [1-t \; \alpha \; x_k]^{-1/\alpha}$.

You can use the generalized Leibnitz formula to relate this back to the multinomial coefficients:

$D_t^m \; \prod_{k=1}^{n+1} g_k(t) = (D_1 + D_2 + \cdots + D_{n+1})^m \; \prod_{k} g_k(t)$

where $D_j$ acts as $d/dt$ only on $g_j(t)$.

The e.g.f. is naturally consistent with the e.g.f. for A094638, which with a change of notation is

$e^{s.(\alpha)x} = (1-\alpha x)^{-1/ \alpha}$,

and is an e.g.f. for plane m-ary trees with $\alpha = m-1$, so

$\prod_k \; (1-\alpha x_kt)^{-1/ \alpha} = \prod_k \; e^{s.(\alpha)x_kt} = \prod_k \; e^{\hat{x}_k t} = e^{tJ.^{\alpha}}$.

Therefore, the discussions in A134264 and the Hirzebruch criterion post below on repeated exponentiaton of an e.g.f., binomial convolutions, and umbral substitution apply to these calculations when $x_k=x$, giving connections among the OEIS entries cited at the top here and the multinomial coefficients.

Taking the log of the e.g.f. gives a relation between the symetric power sum polynomials / functions of the variables / indeterminates and the cumulants formed from the JSPs through A127671, or A263634.

There are several other op reps for the JSPs.

$s_n(\alpha)x^n = \alpha^n \; \frac{(xD+1/\alpha-1)!}{(1/\alpha-1)!}\; x^n$,

so another rep is

$J^\alpha_n = [\prod_{k} \; \frac{(x_k D_{x_k}+1/\alpha-1)!}{(1/\alpha-1)!}\;] \; \alpha^n \; J^0_n$

$= [\prod_{k} \; (x_k D_{x_k})! \; K(1-1/\alpha,1,-:x_kD_{x_k}:)] \; \alpha^n \; J^0_n$,

$= [\prod_{k} \; (x_k D_{x_k})! \; D_{x_k}^{1/\alpha-1}\frac{x_k^{1/\alpha-1}}{(1/\alpha-1)!}] \; \alpha^n \; J^0_n$,

where $K(a,b,x)$ is the Kummer confluent hypergeometric function  (see also earlier posts) and by definition $:xD:^n=x^nD^n$. The reader can use the series expansion for the Kummer function in terms of rising factorials, the transformation $\binom{u-1+n}{n} = (-1)^n \binom{-u}{n}$, and the Chu-Vandermonde identity to confirm that

$(xD_{x})! \; K(1-1/\alpha,1,-:xD_{x}:)] \; x^m = \frac{(m+1/\alpha-1)!}{(1/\alpha-1)!}\; \;x^m$.

Note that $s_n(x)/(1+x)$ are the row polynomials of reversed, unsigned http://oeis.org/A049444 and reversed http://oeis.org/A143491.
Note that $s_n(x)/(1+x)$ are the row polynomials of reversed, unsigned http://oeis.org/A049444 and reversed https://oeis.org/A143491.