×

Uniform asymptotics and zeros of the associated Pollaczek polynomials. (English) Zbl 1462.33003

The associated Pollaczek polynomials \( P_n^{\lambda}(x; a,b,c) \) are defined by the three-term recurrence relation \[(n+c)P_n^{\lambda}(x; a,b,c)= 2[(n-1+\lambda+a+c)z+b]P_{n-1}^{\lambda}(x; a,b,c) \] \[ -(n+2\lambda+c-2)P_{n-2}^{\lambda}(x; a,b,c),\quad n=1,2,\ldots,\] with initial conditions \( P_{-1}^{\lambda}(x; a,b,c)=0 \) and \( P_0^{\lambda}(x; a,b,c)=1. \)
From the abstract: “Two asymptotic approximations are derived for these polynomials; one holds for \( x=1+\frac{t}{n} \) with \( -(a+b)<t \) and \( (a+b)>0, \) and the other holds for \( x=1+\frac{t}{n} \) with \( t \) in a neighborhood of \( t=-(a+b).\) An asymptotic formula is also provided for their largest zeros. ”

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
26C10 Real polynomials: location of zeros
30C10 Polynomials and rational functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI