I have found an approximation for the iterated product:
\( P_b(x) = \prod_{k=1}^{x}sexp_b(k) \) when x is non-integer value.
this product is important because it is resulted in the first derive of \( sexp_b(x) \)
Now! after some logical sequences I made an approximation as below:
Let x = u + v , while u is integer and 0 < v < 1
then the approximation for \( P_b(x) \) is
\( P_b(x) \approx sexp_b(v)^{sexp_b(u)} \prod_{k=1}^{u}sexp_b(k) \)
I can varify it is exactly correct for min v and max v values.
but I can't make sure for v , if v for example equal 0.5 because I don't have Measuring tools or software to compare the variances
I need help.
\( P_b(x) = \prod_{k=1}^{x}sexp_b(k) \) when x is non-integer value.
this product is important because it is resulted in the first derive of \( sexp_b(x) \)
Now! after some logical sequences I made an approximation as below:
Let x = u + v , while u is integer and 0 < v < 1
then the approximation for \( P_b(x) \) is
\( P_b(x) \approx sexp_b(v)^{sexp_b(u)} \prod_{k=1}^{u}sexp_b(k) \)
I can varify it is exactly correct for min v and max v values.
but I can't make sure for v , if v for example equal 0.5 because I don't have Measuring tools or software to compare the variances
I need help.