FoCM 2014 conference

Workshop C5 - Special Functions and Orthogonal Polynomials

December 20, 18:00 ~ 18:30 - Room A21

Quasi-orthogonality of some $_pF_q$ hypergeometric polynomials

University of Pretoria, South Africa   -   kjordaan@up.ac.za

We prove the quasi-orthogonality of some general classes of hypergeometric polynomials of the form $_p F_q\left(\begin{array}{c}-n, \beta_1+k,\alpha_3 \dots, \alpha_p \\ \beta_1, \dots ,\beta_q \end{array};x\right)= \sum_{m=0}^{n}\frac{(-n)_m (\beta_1+k)_m(\alpha_3)_m \dots (\alpha_p)_m}{(\beta_1)_m \dots (\beta_q)_m}\frac{x^m}{m!}$ for $k\in\{1,2,\ldots,n-1\}$ which do not appear in the Askey scheme for hypergeometric orthogonal polynomials. Our results include, as a special case, the order one quasi-orthogonal Sister Celine polynomials $$f_n(a,x)= {}_3F_2 \left(\begin{array}{c}-n,n+1,a\\ \frac{1}{2},1\end{array};x\right)=\sum_{m=0}^{n}\frac{(-n)_m(n+1)_m(a)_m}{\left(\frac 12\right)_m(1)_m}\frac{x^m}{m!}$$ with $a=2$ and $a=3/2$ considered by Dickenson in 1961. The location and interlacing of the real zeros of the quasi-orthogonal polynomials are also discussed.

Joint work with Sarah Jane Johnston (University of South Africa, South Africa).