Summation of Legendre polynomials related to the zeta function

Question

In the wake of my solution to discussion of the problem of summation of Legendre polynomials (https://math.stackexchange.com/questions/2111566/summation-of-legendre-polynomials-sum-l-2-infty-frac2l1l1kl1/2116383#2116383) I stumbled upon the following problem which has interesting aspects.

Consider the sum

$$f_{-2}(z)=\sum _{n=1}^{\infty } \frac{P_n(z)}{n^2}$$

fm2[z_] := Sum[LegendreP[n, z] 1/n^2, {n, 1, \[Infinity]}]

The problem:

1. Find a closed form expression
2. Discuss the functions appearing

Hint: consider the sum

$$g(p,z) = \sum _{n=0}^{\infty } p^n P_n(z)$$

g[p_,z_]:=Sum[LegendreP[n, z] p^n, {n, 0, \[Infinity]}]

EDIT #1

Equivalent formulation of the problem

As shown in §2 of my solution the problem is equivalent to solving the integral

$$fi_{-2}(z) = \int_0^1 \frac{1}{q} \log \left(\frac{2}{\sqrt{q^2-2 q z+1}-q z+1}\right) \, dq$$

for $-1 < z < 1$.

Answer 1

The integral

Integrate[Log[2/(1 - q z + Sqrt[1 + q^2 - 2 q z])]/q, {q, 0, 1}]

may be found in the following way : Calculate the derivative of the integrand with respect to z and determine the antiderivative with respect to q. Then find its limits at q -> 1 (easy) as well as at q -> 0 (harder, use Series[_, {q, 0, 0}]) and integrate their difference then again with respect to z to obtain

(ArcTanh[z]^2 - Log[ (1 - z^2)/4]^2/4 - 
2 (PolyLog[2, Sqrt[(1 - z)/2]] - PolyLog[2, -Sqrt[(1 - z)/2]]))/2 + const

A look at the numerical difference at z = 0 of the original integral and the obtained result shows that const is equal to

(-Pi^2 + 4 ArcSinh[1]^2 + 2 Log[2]^2 + 16 PolyLog[2, 1/Sqrt[2]] + 
4 PolyLog[2, 2 (-1 + Sqrt[2])])/8 = 1.644934066 ... = Pi^2/6.

Answer 2

This will be a step-by-step solution which gives the reader the opportunity to continue at some intermediate result.

§1. First of all we rule out that the problem is trivial by letting Mathematica calculate the infinite sum

Assuming[-1 < z < 1, Sum[LegendreP[n, z] 1/n^2, {n, 1, \[Infinity]}]];

It is returned unevaluated immediately.

Also consulting some of the literature showed no result for this sum. E.g. http://functions.wolfram.com/Polynomials/LegendreP/23/02/, http://www.ucl.ac.uk/~ucahdrb/MATHM242/OutlineCD2.pdf

§2. In order to solve the problem we start with a straightforward approach which, however, leads to an integral which Mathematica cannot solve. This is then the next step of the problem which is herewith also presented for the community to be solved.

Starting from the hint we find the generating function of the Legendre-Polynomials:

g[p_, z_] = Sum[LegendreP[n, z] p^n, {n, 1, \[Infinity]}]

(* Out[1686]= -1 + 1/Sqrt[1 + p^2 - 2 p z] *)

The inverse square under the sum can be generated by twofold Integration thus

Assuming[n > 0, Integrate[1/q Integrate[p^(n - 1), {p, 0, q}], {q, 0, 1}]]

(* Out[1684]= 1/n^2 *)

Now we only have to apply this integration procedure to g.

First the p-integral:

ip = Assuming[{0 < q < 1, 0 < z < 1}, Integrate[1/p g[p, z], {p, 0, q}]]

(* Out[1697]= Log[2/(1 - q z + Sqrt[1 + q^2 - 2 q z])] *)

now the q-integral

ipq = Assuming[{0 < z < 1}, Integrate[1/q ip, {q, 0, 1}]];

$$\int_0^1 \frac{\log \left(\frac{2}{\sqrt{q^2-2 q z+1}-q z+1}\right)}{q} \, dq$$

Unfortunately, this integral is returned unevaluated; also the antiderivative cannot be determined:

Assuming[{0 < z < 1}, Integrate[1/q ip, q]]

(* Out[1699]= \[Integral]Log[2/(1 - q z + Sqrt[1 + q^2 - 2 q z])]/q \[DifferentialD]q *)

And this is just what makes the problem interesting!

We therefore put this integral as an edit into the question as an equivalent problem.

In order to get a feeling for the sum let us first try specific values of z:

fm2m1 = Assuming[{z == -1}, Integrate[1/q ip, {q, 0, 1}]]

(* Out[1710]= -(\[Pi]^2/12) *)

fm20 = Assuming[{z == 0}, Integrate[1/q ip, {q, 0, 1}]]
% // N

(* Out[1714]= 1/6 (-\[Pi]^2 - 3 Log[2]^2 - Log[8] Log[-1 + Sqrt[2]] + 
   3 Log[-1 + Sqrt[2]]^2 - 6 PolyLog[2, 1 - Sqrt[2]] + 
   6 PolyLog[2, 2 - Sqrt[2]]) *)

(* Out[1715]= -0.107492 *)

fm21 = Assuming[{z == 1}, Integrate[1/q ip, {q, 0, 1}]]

(* Out[1712]= \[Pi]^2/6 *)

Numerically the original function is equivalent to

fm2n[z_] := NIntegrate[1/q ip, {q, 0, 1}]

The graph (together with the specific values just calculated) is

Plot[{fm2n[z], fm2m1, fm20, fm21}, {z, -1, 1}, 
 PlotStyle -> {Bold, Dashed, Dashed, Dashed}, 
 PlotLabel -> 
  "The function \!\(\*SubscriptBox[\(f\), \(-2\)]\)(z) \[LongEqual] \
\!\(\*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \
\(\[Infinity]\)]\)\!\(\*FractionBox[TemplateBox[{\"n\",\"z\"},\n\
\"LegendreP\"], SuperscriptBox[\(n\), \(2\)]]\) ", AxesLabel -> {"z", ""}]

EDIT 12.05.17

Unser tangh2014 pointed out in a comment that the sum has a closed form if for $n>1$ we approximate $n^2$ by $n^2-1$ in the denominator.

The approximate sum is then

$$f_{-2a}(z)=\sum _{n=2}^{\infty } \frac{P_n(z)}{n^2-1}+P_1(z)$$

Decomposing the denominator into partial fractions, writing the inverse powers as integrals, using the generating function (now starting at $n=2$), the final integration can be done in closed form with the result

$$f_{-2a}(z)=\frac{1}{4} \left(-2 \sqrt{2-2 z}+z (\log (4)-1)-2 z \log \left(-z+\sqrt{2-2 z}+1\right)-2 \log \left(\frac{z-\sqrt{2-2 z}-1}{z-1}\right)+4\right)$$

The comparison in the plot shows reasonable qualitative agreement

The quatitative agreement is poor close to $z = 0.1$ as can be seen from the graph

Answer 3

This is not a answer how to find, only a solution. From this site closed form expression is:

$\sum _{n=1}^{\infty } \frac{P_n(x)}{n^2}=\frac{1}{24} \left(4 \pi ^2+12 \tanh ^{-1}(x)^2-3 \log ^2\left(\frac{1}{4} \left(1-x^2\right)\right)-48 \text{Li}_2\left(\frac{\sqrt{1-x}}{\sqrt{2}}\right)+12 \text{Li}_2\left(\frac{1-x}{2}\right)\right)$

f[x_, m_] := Sum[LegendreP[n, x]/n^2, {n, 1, m}] // N
Plot[{f[x, 100], 1/24 (4 \[Pi]^2 + 12 ArcTanh[x]^2 - 3 Log[1/4 (1 - x^2)]^2 - 
 48 PolyLog[2, Sqrt[1 - x]/Sqrt[2]] + 
 12 PolyLog[2, (1 - x)/2])}, {x, -1, 1}, PlotStyle -> {Red, {Dashed, Black}}]