# Construct generating functions of series of palindromic polynomials

I have a problem that is generating a series ($d=2,4,\ldots,20,\ldots$) of pairs of $4 d$-degree palindromic (self-reciprocal) polynomials.

The first three members ($d=2,4,6$) of the first pair are: $$\frac{32768}{3} \left(s^8-31 s^6-39 s^4-31 s^2+1\right)$$ and $$\frac{100663296}{5} \left(s^{16}-328 s^{14}+3223 s^{12}-3496 s^{10}+12505 s^8-3496 s^6+3223 s^4-328 s^2+1\right),$$ and $$\frac{85899345920}{21} \left(10 s^{24}-11775 s^{22}+677973 s^{20}-5688979 s^{18}+17249814 s^{16}-39697668 s^{14}+41914740 s^{12}-39697668 s^{10}+17249814 s^8-5688979 s^6+677973 s^4-11775 s^2+10\right).$$ The first three members ($d=2,4,6$) of the second pair (now containing odd powers too of $s$) are $$-\frac{4096}{315} \left(2843 s^8-15360 s^7-22 s^6-25600 s^5-6882 s^4-25600 s^3-22 s^2-15360 s+2843\right)$$ and $$-\frac{2097152 \left(3095503 s^{16}-36126720 s^{15}-441978818 s^{14}+1611939840 s^{13}+47805562 s^{12}-567361536 s^{11}-541456202 s^{10}+5031346176 s^9-2066788010 s^8+5031346176 s^7-541456202 s^6-567361536 s^5+47805562 s^4+1611939840 s^3-441978818 s^2-36126720 s+3095503\right)}{75075}$$ and $$-\frac{2147483648 \left(5660682116 s^{24}-106274488320 s^{23}-3585319303217 s^{22}+21392660889600 s^{21}+110882039596807 s^{20}-397070615445504 s^{19}-229641722669881 s^{18}+1033888042057728 s^{17}+383084268859914 s^{16}-2333132422905856 s^{15}+557684919386502 s^{14}+267997323722752 s^{13}-339034426148082 s^{12}+267997323722752 s^{11}+557684919386502 s^{10}-2333132422905856 s^9+383084268859914 s^8+1033888042057728 s^7-229641722669881 s^6-397070615445504 s^5+110882039596807 s^4+21392660889600 s^3-3585319303217 s^2-106274488320 s+5660682116\right)}{61108047}.$$ I plan to input each of these series to the Mathematica commands FindSequenceFunction and/or FindGeneratingFunction, to discover their underlying rules--but am presently not too optimistic in these regards.

So, is it possible to exploit the palindromic property of these sequences in such a quest? (I, of course, can provide further members of these two sequence.) Are there any standard, recognized families of such polynomials that might be of potential interest?

These polynomials are constituents of an integrand I am seeking to evaluate over $s \in [0,\infty]$ for general integral $d>0$--and pertain to my previous question Extend a two-dimensional hypergeometric-related integration problem by allowing a parameter $d$ to be free. The variables $d$ and $s$ are the same in the two questions. (The members of the pairs are distinguished by the fact that those of the first pair are multiplied by $\log {s}$ in the integrand, and those of the second pair are not.)

In fact, if one chooses to first integrate over $s \in [0,\infty]$, rather than $t \in [0,1]$, one arrives at a "dual" set of pairs of palindromic polynomials. Then, we have the sequence ($d=1,2,3$) of degree $2 d+6$, $$-512 t^8-8192 t^6-18432 t^4-8192 t^2-512$$ and $$16384 t^{14}+802816 t^{12}+7225344 t^{10}+20070400 t^8+20070400 t^6+7225344 t^4+802816 t^2+16384$$ and $$-524288 t^{20}-52428800 t^{18}-1061683200 t^{16}-7549747200 t^{14}-23121100800 t^{12}-33294385152 t^{10}-23121100800 t^8-7549747200 t^6-1061683200 t^4-52428800 t^2-524288$$ and the companion (again $d=1,2,3$) sequence (corresponding to the terms not multiplied by $\log {t}$) $$\frac{640}{3} \left(5 t^8+32 t^6-32 t^2-5\right)$$ and $$-\frac{4096}{35} \left(363 t^{14}+9947 t^{12}+48363 t^{10}+42875 t^8-42875 t^6-48363 t^4-9947 t^2-363\right)$$ and $$\frac{720896}{315} \left(671 t^{20}+41900 t^{18}+564975 t^{16}+2505600 t^{14}+3704400 t^{12}-3704400 t^8-2505600 t^6-564975 t^4-41900 t^2-671\right).$$ (For the first "dual" set ($d=2,4,6$) the odd values ($d=1,3,5$) lead to intractable integrations.)

• A palindromic polynomial of degree $2d$ has form $x^d f(x+x^{-1})$ where $f$ is a polynomial of degree $d$. – Max Alekseyev Aug 25 '17 at 18:54
• Thanks, Max Alekseyev. I've implemented this procedure, and will use it in my search for the rules underlying these palindromic polynomials. – Paul B. Slater Aug 26 '17 at 17:57