Summing Power Series with Polynomial Coefficients

Klippert, John

doi:10.1080/00029890.1983.11971211

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A Sufficient Condition For the Validity Of The Transformatio1

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2005

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“…Interesting notes on transformation of formal series can be found in [8, page 221]. A formula similar to (3.2) with f a polynomial and using finite differences instead of (xD) was presented by Klippert in [18].…”

Section: A Sufficient Condition For the Validity Of The Transformatiomentioning

confidence: 99%

A series transformation formula and related polynomials

Boyadzhiev

2005

International Journal of Mathematics and Mathematical Sciences

106

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We present a formula that turns power series into series of functions. This formula serves two purposes: first, it helps to evaluate some power series in a closed form; second, it transforms certain power series into asymptotic series. For example, we find the asymptotic expansions for ÃŽÂ»>0 of the incomplete gamma function ÃŽÂ³(ÃŽÂ»,x) and of the Lerch transcendent ÃŽÂ¦(x,s,ÃŽÂ»). In one particular case, our formula reduces to a series transformation formula which appears in the works of Ramanujan and is related to the exponential (or Bell) polynomials. Another particular case, based on the geometric series, gives rise to a new class of polynomials called geometric polynomials

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Section: A Sufficient Condition For the Validity Of The Transformatiomentioning

confidence: 99%