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Gamma, Beta, Erf > HarmonicNumber[z] > Introduction to the differentiated gamma functions


Almost simultaneously with the development of the mathematical theory of factorials, binomials, and gamma functions in the 18th century, some mathematicians introduced and studied related special functions that are basically derivatives of the gamma function. These functions appeared in coefficients of the series expansions of the solutions in the logarithmic cases of some important differential equations. They appear in the Bessel differential equation for example. So functions in this group are called the differentiated gamma functions.

The harmonic numbers for integer have a very long history. The famous Pythagoras of Samos (569–475 B.C.) was the first to encounter the harmonic series

in connection with string vibrations and his special interest in music. Richard Suiseth (14th century) and Nicole d’Oresme (1350) studied the harmonic series and discovered that it diverges. Pietro Mengoli (1647) proved the divergence of the harmonic series. Nicolaus Mercator (1668) studied the harmonic series corresponding to the series of and Jacob Bernoulli (1689) again proved the divergence of the harmonic series. The harmonic numbers with integer also appeared in an article of G. W. Leibniz (1673).

In his famous work, J. Stirling (1730) not only found the asymptotic formula for factorial , but used the digamma psi function (related to the harmonic numbers), which is equal to the derivative of the logarithm from the gamma function (). Later L. Euler (1740) also used harmonic numbers and introduced the generalized harmonic numbers .

The digamma function and its derivatives of positive integer orders were widely used in the research of A. M. Legendre (1809), S. Poisson (1811), C. F. Gauss (1810), and others. M. A. Stern (1847) proved the convergence of the Stirling series for the digamma function:

At the end of the 20th century, mathematicians began to investigate extending the function to all complex values of (B. Ross (1974), N. Grossman (1976)). R. W. Gosper (1997) defined and studied the cases and . V. S. Adamchik (1998) suggested the definition for complex using Liouville's fractional integration operator. A natural extension of for the complex order was recently suggested by O. I. Marichev (2001) during the development of subsections with fractional integro‐differentiation for the Wolfram Functions website and the technical computing system Mathematica: