The symmetric Meixner-Pollaczek polynomials with real parameter
The symmetric Meixner-Pollaczek polynomials \(p_n^\lambda(x)\) can be determinated by the generating function \[ {{\exp (x\arctan t)}\over {(1+t^2)^\lambda}}=\sum_{n=0}^\infty p_n^\lambda(x) t^n,\quad \lambda\in\mathcal R. \] It is well known that these polynomials are orthogonal with the weight \[ \omega_\lambda(x)= { 1\over {2\pi}} \biggl| \Gamma \biggl( \lambda+ {{ix}\over 2}\biggr)\biggr| ^2 \] if \(\lambda>0\). In the case where \(\lambda\leq 0\) the Favard Theorem implies that there is no positive real measure \(\mu\) such that \(p_n^\lambda(x)\) are orthogonal with respect to \(\mu\). The author succeeded in proving that \(\{ p_n^\lambda(x)\}^\infty_{n=0}\) is an orthogonal polynomial system for every real \(\lambda\) if we define the associated inner product as follows: \[ \langle f, g\rangle_\lambda=\int_{-\infty}^{+\infty} R^{m_\lambda} (f\cdot g^*)\omega_{\lambda+{{m_\lambda}\over 2}} (x) \,dx \] where \(m_\lambda=\min_{n\in{\mathcal N}}\{ n: \lambda+ {n\over 2} >0\}\), \(R(f(x))={{f(x+i)+f(x-i)}\over 2}\), \(R^n(f)=R(R^{n-1}(f))\), \(g^*\equiv\) the complex conjugate of \(g\) and \(\omega_{\lambda+{{m_\lambda}\over 2}} (x)={1\over{2\pi}} | \Gamma(\lambda+{{m_\lambda}\over 2}+{{ix}\over 2})| ^2\).
scientific article; zbMATH DE number 2162010