New Recursive Representations for the Favard Constants with Application to Multiple Singular Integrals and Summation of Series

Algorithm 1

Mathematica code for exact symbolic and approximate computation with optional 30 digits accuracy of the Favard constants by recursive representations (7a) and (7b).

𝚖=12;   Array  [𝙺,𝚖];  𝙺₁=𝜋2;  𝚍𝚗₋,𝚡₋𝚡∶=^𝚗𝚗!

For  𝚜=1,𝚜≤𝚖/2,𝚜++,

  𝙺_2𝚜+1=12(−1)^𝚜𝚍[2𝚜+1,𝜋]+∑^𝚜_𝚙=1𝙺_{2𝚜+1−2𝚙}(−1)^𝚙−1𝜋^2𝚙(2𝚙)!;

  𝙺_2𝚜=(−1)^𝚜𝚍𝜋2𝚜,2+∑^𝚜_𝚙=1𝙺_{2𝚜+1−2𝚙}(−1)^𝚙−1𝜋2^2𝚙−1(2𝚙−1)!;

For [s = 1,  s ≤ m,  s ++, Print [𝙺_𝚜,  " ",  N[𝙺_𝚜, 30]]];