Absolutely monotonic functions and connection coefficients for polynomials (Q1105066)

The author has shown [same Journal 116, 489-496 (1986; Zbl 0594.26013)] that if \(\{a_ n\}\) is nondecreasing and \((D+a_ n)^ nf(x)\geq 0\) on [a,b] with \(D\equiv d/dx\), then f is analytic on [a,b) if \(\limsup (a_ n/n)<\infty.\) Here he extends this result to a larger class of polynomial differential operators. Much use is made of the connection coefficients that express \((x+a_ n)^ n\) in terms of a given \(\{P_ n(x)\}\), where \(\{a_ n\}\) is nondecreasing. In particular, the quoted theorem persists when \(P_ n(D)f(x)\geq 0,\) provided that the connection coefficients are nonnegative. Applications are made to the cases when \(\{P_ n\}\) is an Appell or Sheffer set, or when \(\{P_ n\}\) satisfies a three-term recurrence relation of the kind satisfied by orthogonal polynomials. An interesting auxiliary result gives necessary and sufficient conditions for an Appell sequence to have all zeros real.