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Logarithmic Binomial Theorem

For all integers n and |x|<a,

lambda_n^((t))(x+a)=sum_(k=0)^infty|_n; k]lambda_(n-k)^((t))(a)x^k,

where lambda_n^((t)) is the harmonic logarithm and |_n; k] is a Roman coefficient. For t=0, the logarithmic binomial theorem reduces to the classical binomial theorem for positive n, since lambda_(n-k)^((0))(a)=a^(n-k) for n>=k, lambda_(n-k)^((0))(a)=0 for n<k, and |_n; k]=(n; k) when n>=k>=0.

Similarly, taking t=1 and n<0 gives the negative binomial series. Roman (1992) gives expressions obtained for the case t=1 and n>=0 which are not obtainable from the binomial theorem.

See also

Harmonic Logarithm, Roman Coefficient

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References

Roman, S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641-648, 1992.

Referenced on Wolfram|Alpha

Logarithmic Binomial Theorem

Cite this as:

Weisstein, Eric W. "Logarithmic Binomial Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogarithmicBinomialTheorem.html

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