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Determinant representations of Appell polynomial sequences. (English) Zbl 1166.15003
The Appell polynomial sequence \((a_n)\), \(n=0,1,2,\dots\), associated with a function \(g\) is defined recursively by differential equations \(a_n'(x)=na_{n-1}(x)\) with initial conditions \(a_n(0)=h^{(n)}(0)\) where \(h=1/g\).
It is proved that
\[ a_n(x)=(-1)^n\frac{n!}{g_0^{n+1}} \begin{vmatrix} 1&g_0&0&0&\dots&0 \\ \frac{x}{1!}&g_1&g_0&0&\dots&0 \\ \frac{x^2}{2!}&g_2&g_1&g_0&\dots&0 \\ \vdots&\vdots&\vdots&\vdots&&\vdots \\ \frac{x^{n-1}}{(n-1)!}&g_{n-1}&g_{n-2}&g_{n-3}&\dots&g_0 \\ \frac{x^n}{n!}&g_n&g_{n-1}&g_{n-2}&\dots&g_1 \end{vmatrix} \]
where \(g_k=\frac{g^{(k)}(0)}{k!}\), \(k=0,1,\dots,n\). As special cases, determinantal representations for Bernoulli and Euler polynomials are obtained.
The proof applies the Leibniz matrix associated with a function \(f\) and a nonnegative integer \(n\), defined by
\[ {\mathcal L}_n(f(x))= \begin{pmatrix} f(x)&0&0&0&\dots&0 \\ \frac{f'(x)}{1!}&f(x)&0&0&\dots&0 \\ \frac{f''(x)}{2!}&\frac{f'(x)}{1!}&f(x)&0&\dots&0 \\ \vdots&\vdots&\vdots&\vdots&&\vdots \\ \frac{f^{(n)}(x)}{n!}&\frac{f^{(n-1)}(x)}{(n-1)!}& \frac{f^{(n-2)}(x)}{(n-2)!}&\dots&\dots&f(x) \end{pmatrix}, \] see D. Kalman and A. Ungar [Am. Math. Mon. 94, 21–35 (1987; Zbl 0622.05002)].

15A15 Determinants, permanents, traces, other special matrix functions
05A40 Umbral calculus
11B68 Bernoulli and Euler numbers and polynomials
33C65 Appell, Horn and Lauricella functions
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