In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. 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 with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.
Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal series. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal series as its series expansion; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative and fractional powers of x are examples of functions that do not have a corresponding formal power series.
Generating functions are not functions in the formal sense of a mapping from a domain to a codomain. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence of term coefficients.

You do not have permission to view the full content of this post. Log in or register now.
  1. D

    Team creates crystals that generate electricity from heat

    To convert heat into electricity, easily accessible materials from harmless raw materials open up new perspectives in the development of safe and inexpensive so-called "thermoelectric materials." A synthetic copper mineral acquires a complex structure and microstructure through simple changes in...
  2. S

    Help Anong tamang payload for tnt smart ?

    Nahihirapan ako sa pagGenerate tsaka po dun sa url host? Pahelp naman sa mga expert creator nang ehi dyan Pasiksik na rin nang lowping at No load Thanks !!
  3. J

    Closed direct link generator

    supported kahit maraming links basta one per line lang Copy and paste the url of mega files you want to download in this site
  4. C

    Closed How to Convert Link into Direct Link

    Yo! Short Tutorial lng.. * COPY url nang mega * Paste sa site na ito ( * Click Generate * Done! Peace Out!
  5. L

    Closed Use at your own risk green packet generator gpz only paki delet if repost

  6. U

    Full Tutorial on Generating Mac Address Effectively using Mac Address Table Part 1

    Hi! just want to share something effective about generating VIP mac address effectively. Download attachment nalang para iwas lechers. ;););):bookworm::bookworm::bookworm: <attachment deleted>