Paper IV: Reciprocal powers and parameter polynomials

Part II Reciprocal coefficients and polynomial families

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7 Reciprocal powers and parameter polynomials

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Reader roadmap

A unit constant term makes the formal reciprocal unambiguous. To compute one coefficient, only finitely many earlier coefficients are needed. The later analytic theory explains how those exact coefficients relate to the full zero set.

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Secant block coefficients 1, 1, 5, 61 and 1385 become ordinary coefficients after division by their matching even factorials. — At \(N=2\) and \(r=1\), the reciprocal is \(\sec z\). The upper row gives block exponential coefficients and the lower row gives ordinary Taylor coefficients. Both rows encode the same five terms.

Since \({\mathsf D}_N(0)=1\), its formal logarithm is defined at the origin. On a simply connected zero free region containing the origin, there is a unique holomorphic branch normalized by \(\log{\mathsf D}_N(0)=0\), and its germ at the origin agrees with the formal logarithm.

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Definition 7.1

For \(r\in{\mathbb C}\), define

\[ {\mathsf S}_N^{(r)}(z) = {\mathsf D}_N(z)^{-r} = \exp\bigl(-r\log{\mathsf D}_N(z)\bigr). \]

Write

\[ {\mathsf S}_N^{(r)}(z) = \sum_{n\geq0}P_{N,n}(r)\frac{z^{Nn}}{(Nn)!}. \]

Define the logarithmic coefficients \(\kappa_{N,m}\) by

\[ -\log\Delta_N(w) = \sum_{m\geq1}\kappa_{N,m}\frac{w^m}{m}. \]

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The polynomial dependence on \(r\) can be read directly from the formal logarithm. We use the complete exponential Bell polynomials \(B_n\) defined by

\[ \exp\left(\sum_{m\geq1}x_m\frac{t^m}{m!}\right) = \sum_{n\geq0}B_n(x_1,\ldots,x_n)\frac{t^n}{n!}. \]

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Theorem 7.2: Bell polynomial formula

For every \(n\geq0\),

\[ P_{N,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\kappa_{N,1}, r\,1!\,\kappa_{N,2}, \ldots, r\,(n-1)!\,\kappa_{N,n} \bigr). \]

For \(n\geq1\), this is a polynomial in \(r\) of exact degree \(n\). Its leading coefficient is

\[ \frac{(Nn)!}{n!(N!)^n}. \]

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Proof of Theorem 7.2

The identity

\[ \Delta_N(w)^{-r} = \exp\left( r\sum_{m\geq1}\kappa_{N,m}\frac{w^m}{m} \right) \]

fits the defining generating function for \(B_n\) with

\[ x_m=r(m-1)!\kappa_{N,m}. \]

The coefficient of \(w^n\) is therefore the displayed Bell polynomial divided by \(n!\). Multiplication by \((Nn)!\) gives \(P_{N,n}(r)\). Since

\[ \Delta_N(w)=1-\frac{w}{N!}+O(w^2), \]

we have \(\kappa_{N,1}=1/N!\). The term \(x_1^n\) gives the stated leading coefficient, which is nonzero.

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The coefficients also satisfy a triangular recurrence that does not require the logarithmic coefficients to be known first.

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Theorem 7.3: Parameter recurrence

We have \(P_{N,0}(r)=1\). For \(n\geq1\),

\[ \begin{aligned} P_{N,n}(r) ={}&- \sum_{k=1}^{n-1} (-1)^k \binom{Nn-1}{Nk} P_{N,n-k}(r) \\ &-r \sum_{k=1}^{n} (-1)^k \binom{Nn-1}{Nk-1} P_{N,n-k}(r). \end{aligned} \]

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Proof of Theorem 7.3

Set \(S={\mathsf D}_N^{-r}\). Logarithmic differentiation gives

\[ {\mathsf D}_NS'+r{\mathsf D}_N'S=0. \]

Taking the coefficient of \(z^{Nn-1}/(Nn-1)!\) gives

\[ \sum_{k=0}^{n-1}(-1)^k \binom{Nn-1}{Nk}P_{N,n-k}(r) +r\sum_{k=1}^{n}(-1)^k \binom{Nn-1}{Nk-1}P_{N,n-k}(r)=0. \]

The term with \(k=0\) in the first sum is \(P_{N,n}(r)\), and every other term has a smaller block index. Rearrangement gives the formula.

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Theorem 7.4: Additive parameter law for reciprocal powers

For all \(r,s\in{\mathbb C}\),

\[ {\mathsf S}_N^{(r)}(z){\mathsf S}_N^{(s)}(z) = {\mathsf S}_N^{(r+s)}(z) \]

as a formal identity and on every common zero-free region with a fixed logarithm. Consequently,

\[ P_{N,n}(r+s) = \sum_{q=0}^{n} \binom{Nn}{Nq} P_{N,q}(r)P_{N,n-q}(s). \]

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Proof of Theorem 7.4

Both statements follow by adding the exponents of the same logarithm and then multiplying the block exponential generating series.

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Corollary 7.5: A sequence of binomial type

Define

\[ A_{N,n}(r) = \frac{n!}{(Nn)!}P_{N,n}(r). \]

Then

\[ A_{N,n}(r+s) = \sum_{q=0}^{n} \binom nq A_{N,q}(r)A_{N,n-q}(s). \]

Thus \((A_{N,n})_{n\geq0}\) is a polynomial sequence of binomial type.

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Proof of Corollary 7.5

Multiply the coefficient identity in Theorem 7.4 by \(n!/(Nn)!\) and simplify the factorials.

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The recurrence in Theorem 7.3 uses the coefficients of the original denominator. A second recurrence uses only the logarithmic coefficients and is often shorter.

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Theorem 7.6: Logarithmic Newton recurrence

Set

\[ b_{N,n}(r)=\frac{P_{N,n}(r)}{(Nn)!}. \]

Then \(b_{N,0}(r)=1\) and for every \(n\geq1\),

\[ nb_{N,n}(r) = r\sum_{k=1}^{n}\kappa_{N,k}b_{N,n-k}(r). \]

Equivalently,

\[ P_{N,n}(r) = \frac r n \sum_{k=1}^{n} \frac{(Nn)!}{(N(n-k))!} \kappa_{N,k}P_{N,n-k}(r). \]

If \(p_k=r\kappa_{N,k}\), then, for \(n\geq1\),

\[ b_{N,n}(r) = \frac1{n!} \det \begin{pmatrix} p_1&-1&0&\cdots&0\\ p_2&p_1&-2&\cdots&0\\ p_3&p_2&p_1&\ddots&\vdots\\ \vdots&\vdots&\ddots&\ddots&-(n-1)\\ p_n&p_{n-1}&\cdots&p_2&p_1 \end{pmatrix}. \]

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Proof of Theorem 7.6

Let

\[ B_r(w)=\Delta_N(w)^{-r} = \sum_{n\geq0}b_{N,n}(r)w^n. \]

Logarithmic differentiation gives

\[ \frac{B_r'(w)}{B_r(w)} = r\sum_{k\geq1}\kappa_{N,k}w^{k-1}. \]

Comparison of the coefficient of \(w^{n-1}\) proves the recurrence.

Let \(D_n\) denote the displayed determinant, with \(D_0=1\). Expansion along the last row gives

\[ D_n = \sum_{k=1}^{n} \frac{(n-1)!}{(n-k)!}p_kD_{n-k}. \]

Here the cofactor sign is cancelled by the product of the negative superdiagonal entries \(-(n-k+1),\ldots,-(n-1)\). Thus \(D_n/n!\) satisfies the same recurrence and initial value as \(b_{N,n}(r)\).

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Corollary 7.7: The first three reciprocal polynomials

For every \(N\geq1\),

\[ P_{N,1}(r)=r, \]

\[ P_{N,2}(r) = r\left[ \frac12\binom{2N}{N}(r+1)-1 \right], \]

and

\[ P_{N,3}(r) = r\left[ \frac{(3N)!}{6(N!)^3}(r+1)(r+2) - \binom{3N}{N}(r+1) +1 \right]. \]

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Proof of Corollary 7.7

The first logarithmic coefficients are

\[ \kappa_{N,1}=\frac1{N!}, \]

\[ \kappa_{N,2}=\frac1{(N!)^2}-\frac2{(2N)!}, \]

and

\[ \kappa_{N,3} = \frac1{(N!)^3} - \frac3{N!(2N)!} + \frac3{(3N)!}. \]

Insert these values into Theorem 7.2 and simplify.

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Corollary 7.8

For \(n\geq1\), the polynomial \(P_{N,n}(r)\) is divisible by \(r\).

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Proof of Corollary 7.8

At \(r=0\), the reciprocal family is the constant function one. Every positive-index block coefficient therefore vanishes.

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Example 7.9

For \(N=1\), one has \({\mathsf D}_1(z)^{-r}=e^{rz}\) and \(P_{1,n}(r)=r^n\). For \(N=2\),

\[ P_{2,1}(r)=r, \]

\[ P_{2,2}(r)=r(3r+2), \]

and

\[ P_{2,3}(r)=r(15r^2+30r+16). \]

At \(r=1\), these are the positive secant coefficients.

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