Generating Function (redirected from Generating functions)

generating function [′jen·ə‚rād·iŋ ‚fəŋk·shən]

(mathematics)

A functiong(x,y) corresponding to a family of orthogonal polynomials ƒ₀(x), ƒ₁(x),…, where a Taylor series expansion ofg(x,y) in powers ofywill have the polynomial ƒ_n(x) as the coefficient for the termyⁿ.

A function,g(y), corresponding to a sequencea₀,a₁, ⋯) whereg(y) =a₀+a₁y+a₂y²+ …. Also known as ordinary generating function.

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Generating Function 

A generating function of the sequence f₀, f₁,..., f_n is the function

assuming that this power series converges for at least one nonzero value of t.

The sequence f₀, f₁,..., f_n,… can be a sequence of numbers or of functions. In the latter case, the generating function depends not only on t but also on the arguments of the functions f_n. For example, if f_n = aqⁿ, where a and q are constants, the generating function is

If the fn are Fibonacci numbers—that is, if f₀ = 0, f₁ = 1, f_n+2 = f_n+1 + f_n—we have

If f_n = T_n (x) are Chebyshev polynomials—that is, if T₀(x) = 1

and T_n (x) = cos (n arc cos x)—then

Knowledge of the generating function of a sequence often makes it easier to study the properties of the sequence. Generating functions are used in probability theory, in the theory of functions, and in the theory of invariants in algebra. Methods involving generating functions were first applied by P. Laplace to solve certain problems in probability theory.

REFERENCES

Feller, W. Vvedenie v teoriiu veroiatnostei i eeprilozheniia, 2nd ed., vols. 1–2. Moscow, 1967. (Translated from English.)
Natanson, I. P. Konstruktivnaia teoriia funktsii. Moscow-Leningrad, 1949.