Seminar Series on Operator Algebras and Related Fields

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\title{\textbf{Parametric Appell Sequences: Differential and Combinatorial Structures}}
\author{ {\large \textbf{rabeh jakdali$^{1}$,Laala Khaldi$^{2}$}}\[0.3em]
%\textsuperscript{} \and %Research Partner\textsuperscript{2}\
\small \textsuperscript{1,2} LIM Laboratory, Department of Mathematics
, Faculty of Exact Sciences
,\ University of Bouira, 10000 Bouira, Algeria\dots}
\date{(27–29 nov. 2026)}

\begin{document}
\maketitle

\begin{abstract}
\noindent This paper introduces unconventional algebraic perspectives on Bernoulli and Euler polynomials. By exploiting their properties as Appell polynomial sequences and utilizing their classical generating functions, we establish the existence of novel parametric polynomial families. As a major novel contribution, we prove that these parametric families satisfy elegant second-order linear differential recurrences. Furthermore, we derive an explicit formula for Euler polynomials in terms of Stirling numbers of the second kind, bridging continuous operations with discrete combinatorial structures.
\end{abstract}
\vspace{0.5cm}
\section{Introduction}
Bernoulli polynomials $B_n(x)$ and Euler polynomials $E_n(x)$ have been subjects of profound interest in mathematical analysis. Both families belong to the class of Appell polynomial sequences, satisfying the differential property $A_{n}^{\prime}(x) = nA_{n-1}(x)$.

In this presentation, we divert from traditional recurrence relations to explore an unconventional structural property: representing these polynomials through a transformed variable $t = \frac{1}{2}x(x-1)$. We expand upon existing literature by demonstrating that this transformation leads to a set of novel second-order differential recurrence relations.

\section{Parametric Families and Combinatorial Links}
\begin{theorem}
There exist families of polynomials with complex coefficients $b_n(x)$ and $e_n(x)$ such that for any integer $n \ge 1$, $deg(b_n(x)) = deg(e_n(x)) = n$, satisfying:
\begin{align}
B_{2n}(x) - B_{2n} &= 2n b_n\left(\frac{1}{2}x(x-1)\right), \
E_{2n}(x) &= 2n e_n\left(\frac{1}{2}x(x-1)\right).
\end{align}
\end{theorem}

\begin{proof}
Setting $y = \frac{1}{2}$ and $z = \frac{2x-1}{2}$, we expand the Appell translation formulas using the binomial sum. Applying the algebraic identity $(2x-1)^2 = 8\left(\frac{x(x-1)}{2}\right) + 1$, we substitute the squared terms with a new variable $t = \frac{1}{2}x(x-1)$, which factors the resulting sums into the explicitly constructed polynomials $b_n(t)$ and $e_n(t)$.
\end{proof}

\begin{theorem}
For $m \ge 0$, the Euler polynomials can be explicitly formulated as:
\begin{equation}
E_{m}(x) = \sum_{k=0}^{m} \sum_{j=1}^{k+1} \binom{m}{k} \frac{(-1)^{j+k-1}(j-1)!}{2^{j+1}} S(k+1, j) x^{m-k},
\end{equation}
where $S(k, j)$ denotes the Stirling numbers of the second kind.
\end{theorem}
\begin{proof}
We begin by differentiating both sides of the generating function of Euler polynomials, $\sum_{n=0}^\infty E_n(x) \frac{z^n}{n!} = \frac{2e^{xz}}{e^z+1}$, with respect to $z$, exactly $m$ times. Applying the generalized Leibniz formula, we obtain:
$$ \sum_{n=m}^{\infty}E_{n}(x)\frac{z^{n-m}}{(n-m)!} = 2\sum_{k=0}^{m}\binom{m}{k}(e^{xz})^{(m-k)}\left(\frac{1}{e^z+1}\right)^{(k)} .$$ We now utilize the established identity for the $k$-th derivative of $(1+e^z)^{-1}$, which is explicitly expressed in terms of Stirling numbers of the second kind, $S(k+1,j)$: $$ \left(\frac{1}{1+e^z}\right)^{(k)} = (-1)^{k}\sum_{j=1}^{k+1}\frac{(-1)^{j-1}(j-1)!}{(1+e^z)^{j}}S(k+1,j). $$ Substituting this identity back into our Leibniz expansion and taking the limit as $z \to 0$, the sum on the left side collapses to $E_{m}(x)$. On the right side, the limits evaluate to $(e^{xz})^{(m-k)} \to x^{m-k}$ and $(1+e^z)^j \to 2^j$, which immediately yields the desired explicit formula: $$ E_{m}(x) = \sum_{k=0}^{m}\sum_{j=1}^{k+1}\binom{m}{k}\frac{(-1)^{j+k-1}(j-1)!}{2^{j+1}}S(k+1,j)x^{m-k} .$$ This completes the proof. \end{proof} \section{Our Extension: Differential Recurrence Relations} To establish a deeper intrinsic property of the parametric families, we introduce a novel differential operator perspective for the variable $t = \frac{1}{2}x(x-1)$. \begin{theorem} The parametric Euler polynomials $e_n(t)$ satisfy the following second-order linear differential recurrence relation for $n \ge 2$: \begin{equation} \left(2t + \frac{1}{4}\right) e''_n(t) + e'_n(t) = (2n-1)(2n-2) e_{n-1}(t). \end{equation} Similarly, the parametric Bernoulli polynomials $b_n(t)$ satisfy: \begin{equation} \left(2t + \frac{1}{4}\right) b''_n(t) + b'_n(t) = (2n-1) \left[ (2n-2)b_{n-1}(t) + B_{2n-2} \right]. \end{equation} \end{theorem} \begin{proof} Let $t = \frac{1}{2}x(x-1)$, which gives $\frac{dt}{dx} = x - \frac{1}{2}$. From Theorem 1, we have $E_{2n}(x) = 2n e_n(t)$. Applying the chain rule to differentiate with respect to $x$: $$ E'_{2n}(x) = 2n e'_n(t) \left(x - \frac{1}{2}\right). $$ Differentiating a second time yields: $$ E''_{2n}(x) = 2n \left[ e''_n(t) \left(x - \frac{1}{2}\right)^2 + e'_n(t) \cdot 1 \right] .$$ Notice that $\left(x - \frac{1}{2}\right)^2 = x^2 - x + \frac{1}{4} = 2t + \frac{1}{4}$. Substituting this back, we get: $$ E''_{2n}(x) = 2n \left[ \left(2t + \frac{1}{4}\right) e''_n(t) + e'_n(t) \right] .$$ By the Appell property, we also know that $E''{2n}(x) = 2n(2n-1)E{2n-2}(x)$. Using Theorem 1 again, $E_{2n-2}(x) = (2n-2)e_{n-1}(t)$. Equating the two expressions for $E''{2n}(x)$ gives: $$ 2n \left[ \left(2t + \frac{1}{4}\right) e''_n(t) + e'_n(t) \right] = 2n(2n-1)(2n-2)e_{n-1}(t). $$ Dividing by $2n$ completes the proof for $e_n(t)$. An identical approach using $B''{2n}(x) = 2n(2n-1)B_{2n-2}(x)$ proves the relation for $b_n(t)$. \end{proof} \begin{remark} This new differential identity demonstrates that the transformed polynomials $e_n(t)$ and $b_n(t)$ are not merely algebraic simplifications, but they inherently map the Appell sequence into a space governed by second-order differential operators, presenting a new pathway for fractional integro-differential exploration.
\end{remark}
\begin{thebibliography}{9}
\bibitem{Khaldi2024}
L. Khaldi and R. Boumahdi, \emph{On Some Properties of Bernoulli and Euler Polynomials}, Palestine Journal of Mathematics, Vol 13(4) (2024), 1037-1043.
\end{thebibliography}
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