# Identify power series with coeffs. that are palindromic polynomials of a param.

1. Jul 13, 2012

### jsm10

Can someone help me to identify the type of power series for which the coefficients are palindromic polynomials of a parameter?

More specifically, for a particular function f(x;a) with x, a, and f() in ℝ1, a > 0, and an exponential power series representation

F0 + F1x/1! + F2x2/2! + F3x3/3! +...

each of the coefficients, F1, F2, F3, ..., is a palindromic polynomial in a (e.g., F1 = a + 1, F2 = 2a2 - a + 2, F3 = 111a3 - 142a2 - 142a + 111).

The roots of each polynomial are about equally divided between complex conjugate pairs and real reciprocal pairs. For a polynomial of odd degree, -1 is always a root.
None of the roots of successive polynomials is the same, so I think the series cannot be hypergeometric or q-hypergeometric. The real roots of each of the polynomials are positive (except the single -1 for each odd-degree polynomial), and they appear to converge to an integer value for successive polynomials (e.g., for F3, F4, ..., F12,
the largest real root is 1.6862, 2.7565, 3.2367, 3.4961, 3.6493, 3.7457, 3.8096, 3.8850, 3.9079, 3.9251; for a different f(), the roots converge to 9).

Any information about such a power series would be greatly appreciated.

2. Jul 13, 2012

### haruspex

If you substitute b = √a, y = bx and write as a power series in y, the coefficient polynomials take the form Ʃcn(bn+b-n), which suggests hyperbolic trig functions.