PROPERTIES OF LEGENDRE EXPANSIONS RELATED TO SERIES OF STIELTJES AND APPLICATIONS TO PI-PI SCATTERING

Legendre expansions whose coefficients are those of a series of Stieltjes are considered. It is shown that the analyticity domain of a function defined by such an expansion is the cut plane and that sequences of approximants may be defined which converge to the function in this domain, with each approximant determined from a finite number of coefficients in the expansion. These approximants are related to the Padé approximants of the corresponding series of Stieltjes. It is shown that if the coefficients satisfy a « Froissart-Gribov »-type representation with positive weight, then they are also coefficients of series of Stieltjes. It follows that the above results may be applied to the π-π scattering amplitude A(s, t) for certain states when 0≤s<4. In particular the approximation of A(s, t) is the complex t-plane, when only the first few partial waves a l(s) are known, is discussed and the interpolation of the a l(s) for noninteger l is also considered. Another consequence is that the a l(s) satisfy an infinite set of determinantal inequalities when 0≤s<4. © 1969 Società Italiana di Fisica.