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can be proved by the techniques at Stirling numbers and exponential generating functions#Stirling numbers of the first kind and Binomial coefficient#Ordinary generating functions. The table in section 6.1 of Concrete Mathematics provides a plethora of generalized forms of finite sums involving the Stirling numbers. Several particular finite ...
There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease ...
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:
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]
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
By this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1) H 0 = 0, (2) H x = H x−1 + 1/x for all complex numbers x except the non-positive integers, and (3) lim m→+∞ (H m+x − H m) = 0 for all complex values x.
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.
The Pell numbers are defined by the recurrence relation: = {=; =; + In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number, plus the Pell number before that.