New Inverse Series Relations for Finite and Infinite Series with Applications

Some new series inversion formulas of the general form F(n)=sum form k=0 to r(A_(k,m)f(n-mk)) if and only if f(n)=sum form k=0 to r(B_(k,n)F(r_o-mk)) valid for either r=n/m]or r=∞ are presented. These relations generalize many of those given by the author in a long series of preceding papers. An interesting example is given by A_(k,n)=(-y)~kA_k(p-λn,t(1+λm)) and B_(k,n)=y~kA_k(p-λn,(1-t)(1+λm)) where A_k(a,b)=a/(a+bk) in terms of binomial coefficients. Here p,t,y and λ are arbitrary complex numbers. A corresponding Abel coefficient case occurs which uses numbers of the form a(a+bi)(i-1)/i!. An application to special functions studied by Singhal and Kumari is given, and it is also shown that sum form k=0 to ∞(z~kA_k(a+ck,b))=x~a(x-b(x-1))/(x-(b+c)(x-1)), where z=(x-1)x~(-b-c), with a corresponding case for the Abel coefficients sum from k=0 to ∞(z~kB_k(a+ck,b))=x~o(1-b logx)/(1-(b+c)log x),where z=(log x)x~(-b-c) From these expansions we then have easily the new convolution formula for Rothe coeffici