On a Function Which Occurs in the Theory of the Structure of Polymers

plays an essential role in his researches on the theory of the structure of polymers. In this context s is a rational number, such as 3, -2 or I and x is a real number between 0 and 1. While the series (1) is rapidly convergent near 0, near 1 it is so slowly convergent as to be quite useless for computation, and Mr. Jacobson asked me to find some other representation for? (x, s) which is amenable to numerical evaluation when x is only slightly less than 1. In response to his request I discovered the expansions (9) and (10), which it is the purpose of the present paper to derive, along with some other properties of? (x, s) which seem of possible interest though not useful in M. Jacobson's problems. I have also computed a table of the values of? (x, s) when 0 1 does the series remain convergent at the point x = 1; then