gamma(x) + sin(x * pi) agrees with the factorial [shifted by one, because the gamma function has that stupid shift in its definition for no good reason] on inte

gamma(x) + sin(x * pi) agrees with the factorial [shifted by one, because the gamma function has that stupid shift in its definition for no good reason] on integer arguments, but does not satisfy the recurrence relation on non-integer arguments. That having been said, one can devise infinitely many analytic functions which do satisfy the recurrence relation in general. See my other comment.