How to quickly calculate the following function？

Hi all,I want to calculate the function shown below:

This is Lerch transcendent function( https://en.wikipedia.org/wiki/Lerch_zeta_function ).

I use the following formula to calculate it, but this is too slow (I need to calculate it many times)：

There is also no official function here to calculate it.

So I would like to know how to calculate it quickly, thank you all in advance！

[KSSV]

Show us your code...

[ma Jack]

function [out] = Lerch_fun(z,s,v)
%Lerch_fun(exp(1j*2),3,0.1)
% Lerch transcendent
%\Phi(z, s, v)=\sum_{n=0}^{\infty} \frac{z^{n}}{(n+v)^{s}}
%syms t
f=@(t)((t.^(s-1)).*exp(-v*t))./(1-z.*exp(-t));
out=(1/gamma(s))*(integral(f,0,Inf));

end

[Walter Roberson]

For your purposes, will any of the parameters be constant ?

I am wondering whether there might be efficiency improvements available for the case of one of the parameters being constant.

[ma Jack]

They are all variables.

[Walter Roberson]

https://mathoverflow.net/questions/335620/what-is-the-current-fastest-method-to-calculate-lerchs-phi-transcendent

Answer 1

[Jan]

See also: https://fredrikj.net/blog/2022/02/computing-the-lerch-transcendent/

Calculating the sum seems to be more stable. For cases, in which the integral method is successful, the sum is 10 to 200 times faster for the given test data:

function test_Lerch
% See: https://people.math.sc.edu/Burkardt/py_src/polpak/lerch_values.py
% and: https://people.sc.fsu.edu/~jburkardt/m_src/test_values/test_values.html

a_vec = [0.0E+00, ...
   0.0E+00, ...
   0.0E+00, ...
   1.0E+00, ...
   1.0E+00, ...
   1.0E+00, ...
   2.0E+00, ...
   2.0E+00, ...
   2.0E+00, ...
   3.0E+00, ...
   3.0E+00, ...
   3.0E+00];

s_vec = [2, 3, 10, ...
   2, 3, 10, ...
   2, 3, 10, ...
   2, 3, 10];

z_vec = [0.1000000000000000E+01, ...
   0.1000000000000000E+01, ...
   0.1000000000000000E+01, ...
   0.5000000000000000E+00, ...
   0.5000000000000000E+00, ...
   0.5000000000000000E+00, ...
   0.3333333333333333E+00, ...
   0.3333333333333333E+00, ...
   0.3333333333333333E+00, ...
   0.1000000000000000E+00, ...
   0.1000000000000000E+00, ...
   0.1000000000000000E+00];

f_vec = [ ...
   0.1644934066848226E+01, ...
   0.1202056903159594E+01, ...
   0.1000994575127818E+01, ...
   0.1164481052930025E+01, ...
   0.1074426387216080E+01, ...
   0.1000492641212014E+01, ...
   0.2959190697935714E+00, ...
   0.1394507503935608E+00, ...
   0.9823175058446061E-03, ...
   0.1177910993911311E+00, ...
   0.3868447922298962E-01, ...
   0.1703149614186634E-04];

for k = 1:numel(z_vec)
   z = z_vec(k);
   s = s_vec(k);
   a = a_vec(k);
   f = f_vec(k);
   
   fprintf('\nk: %d\n', k)
   tic
   for k = 1:1
      y1 = Lerch_fun_integral(z,s,a);
   end
   toc
   
   tic
   for k = 1:1
      y2 = Lerch_fun_sum(z,s,a);
   end
   toc
   
   fprintf('Ref: %.16g\n', f);
   fprintf('Int: %.16g  delta: %16g\n', y1, abs(y1 - f));
   fprintf('Sum: %.16g  delta: %16g\n', y2, abs(y2 - f));
end

end

function out = Lerch_fun_integral(z,s,a)
f=@(t)((t.^(s-1)).*exp(-a*t))./(1-z.*exp(-t));
out=(1/gamma(s))*(integral(f,0,Inf));
end


function out = Lerch_fun_sum(z, s, a)
out = 0;

if z <= 0
   return
end

lim = 1e-16;
k   = 0;
z_k = 1;

term = Inf;

while abs(term) > lim * (1.0 + abs(out))
   if a + k ~= 0
      term = z_k / (a + k)^s;
      out  = out + term;
   end
   k   = k + 1;
   z_k = z_k * z;
end
end