Ordinary generating functions definition of Ordinary generating functions in the Free Online Encyclopedia.

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.

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.

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ordinary bicycle
Ordinary Break
Ordinary Cadet
Ordinary Capital Adequacy Requirement
ordinary care
ordinary care
ordinary care
Ordinary Circular Slot
Ordinary Circular Slot Antenna
Ordinary Common Sense
Ordinary Concentrically Braced Frames
ordinary construction
Ordinary Council Meeting
ordinary course
Ordinary course of business
Ordinary course of business
ordinary cut
Ordinary Decent Criminal
Ordinary derivative
Ordinary derivative
Ordinary derivative
Ordinary derivative
Ordinary derivative
Ordinary differential equation
Ordinary differential equation
Ordinary differential equations
Ordinary Dividend
Ordinary Dividends
ordinary gear train
Ordinary generating function

Ordinary generating functions

Ordinary Grade
ordinary grades
Ordinary High Water
Ordinary High Water Level
Ordinary high water mark
Ordinary high water mark
Ordinary High-Water Line
Ordinary income
Ordinary income
Ordinary Income or Loss
Ordinary Incomes
ordinary index
Ordinary interest
ordinary judgment
Ordinary language philosophy
Ordinary language philosophy
Ordinary language philosophy
ordinary lay
Ordinary least squares
Ordinary least squares
Ordinary least squares
Ordinary Least Squares Method
Ordinary least squares regression
Ordinary least squares regression
Ordinary least squares regression
Ordinary least squares regression
Ordinary least squares regression
Ordinary Least-Squares Estimation
Ordinary Leave
Ordinary Level

Generating Function (redirected from Ordinary generating functions)

REFERENCES

Generating Function
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