Zeros of linear combinations of Laguerre polynomials

Abstract

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Laguerre polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_j\tilde L^\alpha_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\cdots ,K$, are real numbers with $\gamma_0=1$, $\gamma_K\not =0$. We consider four different normalizations of Laguerre polynomials: the monic Laguerre polynomials $\hat L_n^\alpha$, the polynomials $\mathcal L_n^\alpha=n!L_n^\alpha/(1+\alpha)_n$ (so that $\mathcal L_n^\alpha(0)=1$), the standard Laguerre polynomials $(L_n^\alpha)_n$ and the Brenke normalization $L_n^\alpha/(1+\alpha)_n$. We show the key role played by the polynomials $Q(x)=\sum_{j=0}^K(-1)^j\gamma_j(x)_{K-j}$ and $P(x)=\sum_{j=0}^K\gamma_jx^{K-j}$ to solve this problem: $Q$ in the first case and $P$ in the second, third and forth cases. In particular, in the first case, if all the zeros of the polynomial $Q$ are real and less than $\alpha+1$, then all the zeros of $q_n$, $n\ge K$, are positive. In the other cases, if all the zeros of $P$ are real then all the zeros of $q_n$, $n\ge K$, are also real. If $P$ has $m>1$ non-real zeros, there are important differences between the four cases. For instance in the first case, $q_n$ has still only real zeros for $n$ big enough, but in the fourth case $q_n$ has exactly $m$ non-real zeros for $n$ big enough.