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In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
where γ m are the Stieltjes constants and δ m,0 represents the Kronecker delta function. Notice that this last identity immediately implies relations between the polylogarithm functions, the Stirling number exponential generating functions given above, and the Stirling-number-based power series for the generalized Nielsen polylogarithm functions.
For a fixed integer n, the ordinary generating function for Stirling numbers of the second kind {}, {}, … is given by = {} = (), where () are Touchard polynomials. If one sums the Stirling numbers against the falling factorial instead, one can show the following identities, among others:
The mathematical motivation for this type of notation, as well as additional Stirling number formulae, may be found on the page for Stirling numbers and exponential generating functions. Another infrequent notation is s 1 ( n , k ) {\displaystyle s_{1}(n,k)} and s 2 ( n , k ) {\displaystyle s_{2}(n,k)} .
The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences.
An alternative method for deriving the same generating function uses the recurrence relation for the Bell numbers in terms of binomial coefficients to show that the exponential generating function satisfies the differential equation ′ = (). The function itself can be found by solving this equation. [11] [12] [13]
One possible generating function for such partitions, taking k fixed and n variable, is = =. More generally, if T is a set of positive integers then the number of partitions of n, all of whose parts belong to T, has generating function
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.