Paper IV: Logarithmic derivatives and Riccati equations

8 Logarithmic derivatives and Riccati equations

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A coefficient calculation at order two

Write \(\Delta_2(w)^{-r}=\sum_{n\geq0}b_n(r)w^n\). Since

\[ \Delta_2(w)=1-\frac w2+\frac{w^2}{24}+O(w^3), \qquad -\log\Delta_2(w)=\frac w2+\frac{w^2}{12}+O(w^3), \]

the first logarithmic coefficients are \(\kappa_{2,1}=1/2\) and \(\kappa_{2,2}=1/6\). Newton's recurrence gives

\[ b_1(r)=\frac r2,\qquad 2b_2(r)=r\left(\frac r4+\frac16\right). \]

Restoring the factorial normalization yields \(P_{2,1}(r)=r\) and \(P_{2,2}(r)=r(3r+2)\). At \(r=1\), the second value is five. No zero-location assumption is needed for this finite calculation.

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On reciprocal orders zero through two, the first two block coefficients are r and r times (three r plus two), with marked values one and five at r equals one. — The two exact parameter polynomials from the worked calculation. Each plotted curve evaluates its displayed formula. The points at \(r=1\) recover the first two positive-index secant coefficients.

Definition 8.1

Define the transfer tangent

\[ {\mathsf T}_N(z) =-\frac{{\mathsf D}_N'(z)}{{\mathsf D}_N(z)}. \]

Near the origin, its convergent block expansion has the form

\[ {\mathsf T}_N(z) = \sum_{n\geq0}T_N(n) \frac{z^{Nn+N-1}}{(Nn+N-1)!}. \]

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Proposition 8.2: Logarithmic coefficient formula

The transfer tangent satisfies

\[ {\mathsf T}_N(z) = N\sum_{m\geq1}\kappa_{N,m}z^{Nm-1}. \]

Equivalently,

\[ T_N(n) = N(Nn+N-1)!\kappa_{N,n+1}. \]

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Proof of Proposition 8.2

Since \({\mathsf D}_N(z)=\Delta_N(z^N)\),

\[ -\frac{{\mathsf D}_N'(z)}{{\mathsf D}_N(z)} = -Nz^{N-1}\frac{\Delta_N'(z^N)}{\Delta_N(z^N)}. \]

Differentiate the defining series for \(-\log\Delta_N\) and compare coefficients.

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Theorem 8.3: Tangent recurrence

The coefficients satisfy \(T_N(0)=1\) and

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

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Proof of Theorem 8.3

The defining equation is

\[ {\mathsf D}_N{\mathsf T}_N=-{\mathsf D}_N'. \]

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

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

Isolating the \(k=0\) term gives the recurrence.

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Corollary 8.4: Derivative of the reciprocal family

For every \(r\in{\mathbb C}\), the formal identity

\[ \frac{d}{dz}{\mathsf S}_N^{(r)}(z) = r{\mathsf T}_N(z){\mathsf S}_N^{(r)}(z) \]

holds. The same identity holds analytically on every zero-free domain on which the logarithm used to define \({\mathsf S}_N^{(r)}\) is fixed.

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Proof of Corollary 8.4

Differentiate \({\mathsf D}_N^{-r}\) on a zero-free region. The formal identity follows from the same logarithmic calculation.

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For a function \(g\), define the complete differential Bell polynomial \(Y_N\) by

\[ \frac{d^N}{dz^N}e^{g(z)} = e^{g(z)}Y_N\bigl(g',g'',\ldots,g^{(N)}\bigr). \]

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Theorem 8.5: Scalar higher Riccati identity

At every point of the open set \(\{z:{\mathsf D}_N(z)\ne0\}\),

\[ Y_N\bigl( -{\mathsf T}_N, -{\mathsf T}_N', \ldots, -{\mathsf T}_N^{(N-1)} \bigr) =-1. \]

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Proof of Theorem 8.5

Write \({\mathsf D}_N=e^g\) locally. Then \(g'=-{\mathsf T}_N\). Divide the equation \({\mathsf D}_N^{(N)}=-{\mathsf D}_N\) by \({\mathsf D}_N\) and use the defining identity for \(Y_N\).

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This scalar identity follows from one ordinary differential equation. It does not redevelop the general nonlinear operation-tree theory of Paper III.

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Example 8.6

For \(N=1\), \({\mathsf T}_1=1\). For \(N=2\), \({\mathsf D}_2=\cos z\) and \({\mathsf T}_2=\tan z\). The Riccati equation becomes

\[ {\mathsf T}_2'=1+{\mathsf T}_2^2. \]

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