Chebyshev Expansions and Rational Approximations for Some Special Functions and Analytic Continuation Formulas for These Special Functions.

摘要

Let A(z) = (A sub m)(z) + (a sub m)(z sup m)B(z,m) where (A sub m)(z) is a polynomial in z of degree m-1. Suppose A(z) and B(z,m) are approximated by main diagonal Pade approximations of order n and r respectively. Suppose that the number of operations needed to evaluate both sides of the above equations by means of the Pade approximations and polynomial noted are the same. Thus 4n = 3m + 4r. We address ourselves to the question of which procedure is more efficient. That is, which procedure produces the smallest error. A variant of this problem is the situation where A(z) and B(z,m) are approximated by their representations in infinite series of Chebyshev polynomials of the first kind truncated after n and r terms respectively. Here n = m+r.