Can someone help me to identify the type of power series for which the coefficients are palindromic polynomials of a parameter?(adsbygoogle = window.adsbygoogle || []).push({});

More specifically, for a particular functionf(x;a) withx,a, andf() in ℝ^{1},a> 0, and an exponential power series representation

F_{0}+F_{1}x/1! +F_{2}x^{2}/2! +F_{3}x^{3}/3! +...

each of the coefficients,F_{1},F_{2},F_{3}, ..., is a palindromic polynomial ina(e.g.,F_{1}=a+ 1,F_{2}= 2a^{2}-a+ 2,F_{3}= 111a^{3}- 142a^{2}- 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., forF_{3},F_{4}, ...,F_{12},

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 differentf(), the roots converge to 9).

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

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

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