Elephant polynomials

Abstract

In this note, we study a family of polynomials that appear naturally when analysing the characteristic functions of the one-dimensional elephant random walk. These polynomials depend on a memory parameter $p$ attached to the model. For certain values of $p$, these polynomials specialise to classical polynomials, such as the Chebychev polynomials in the simplest case, or generating polynomials of various combinatorial triangular arrays (e.g.\ Eulerian numbers). Although these polynomials are generically non-orthogonal (except for $p=\frac{1}{2}$ and $p=1$), they have interlacing roots. Finally, we relate some algebraic properties of these polynomials to the probabilistic behaviour of the elephant random walk. Our methods are reminiscent of classical orthogonal polynomial theory and are elementary.

Figures (2)

• Figure 1: According to Proposition \ref{['prop:interlacing']}, the first six polynomials $R_1(x),\ldots,R_6(x)$ have interlaced roots. Top left display: $a=\frac{1}{4}$; top right: $a=\frac{1}{2}$, these are the Chebychev polynomials; bottom left: $a=\frac{3}{2}$; bottom right: $a=+\infty$.
• Figure 2: From top left to bottom right, the first six polynomials $S_1(x),\ldots,S_6(x)$ for $a=-\frac{1}{2},-1,-\frac{3}{2},-\infty$ have interlaced roots, according to Proposition \ref{['prop:interlacing_imaginary']} (except $S_2(x)=-1$ in the case $a=-1$)

Theorems & Definitions (22)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 5
• proof
• proof : Proof of the identity \ref{['eq:recurrence_varphi']}
• proof : Proof of Proposition \ref{['prop:interlacing']}
• proof : Proof of Proposition \ref{['prop:interlacing_imaginary']}
• proof : Proof of Proposition \ref{['prop:non_orthogonal']}