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. P

    Generate Code via Prompts

    Hidden content
  2. R

    Help Pa generate img. sa Midjourney

    Pa generate po ng image sa Midjourney...bale ito po yung prompt: Collage showing a safe interaction in the digital world.
  3. B


    AI that can be used for cartoonish, hair design, Icon Generate, and SDXL Turbo for FREE and UNLIMITED. Hidden content
  4. P

    Anong ai app po ba ang kayang mag generate neto

    tanong ko lang po kung anong ai app kaya eto. #photonotmine credit to the owner
  5. C

    Help Generate a code

    Generate a code that search for latest recent post on facebook based on keywords or query. the results will be generated as an rss or atom file
  6. Z

    Help Anong best AI to generate logos?

    Mga paps! Meron ba kayong alam na AI sites to generate logos using prompts?
  7. A

    Help Generate EMEI Token for Realme

    May marunong/alam ba dito paano makapag generate? Nakalock pala kasi tong sa Realme kaya di ko magamit yung ##3646633## para makapag change ng band 🥲
  8. V

    Help Email

    How to create unlimited email po?
  9. I


  10. S

    A.I generate a PowerPoint

    May a.i ba na pwedi Maka generate Ng power power point bukod sa may nalalaman paba kayung ibang a.i na makapag generate Ng power point.
  11. N

    Question --RE: REACT-LEAFLET (mag generate ng MAP IMAGE)

    Good day po matatanong lang po sana ako regarding po sa REACT-LEAFLET.. YUNG map po kasi na galing sa MapContainer po gusto ko po gawin IMAGE sya tapos ipi-print ko po sa pdf .. alam ko na po yung sa PRINTING SA PDF , pero yung GAGAWING IMAGE PO SI MapContainer HINDI KO PO ALAM gawin.. baka...
  12. B

    Here are some Useful ai website that generate images through text part #5

    1. uses deep neural networks to transform your text descriptions into visual artwork. The platform employs style transfer algorithms to apply the characteristics of famous artists or styles to your input text, resulting in unique and personalized images. It provides an...
  13. B

    Useful ai website part #4 generate image through text

    1. VQ-VAE-2 by DeepMind: VQ-VAE-2 is an AI model developed by DeepMind that can generate high-quality images from textual descriptions. It uses vector quantization and variational autoencoders to produce coherent and realistic images based on given prompts. 2. PaintsChainer: PaintsChainer is an...
  14. S

    Help Qr Code Library for scanning using phone and generate QR

    Baka may alam po kayo diyang libraries tapos po yung pamg scan ng Qr is responsive pa din sa phone pag phone ginamit pang scan using Php, JavaScript, ajax, etc. Na pwede gamitin sa system
  15. N


    How to generate a fake photo? If you want to register in any social network, but do not want to betray your real face, specially created neural networks will help you. The service allows the user to generate a random photo of a person who simply does not...
  16. N

    Generate token realme

    Pa help generate token ang realme 8 5G ko para lang sa band mode please
  17. P

    Help QR Generate With Editable Link and Image.

    Hi mga ka-PHC meron ba kayo alam na QR site na pwede mag create ng QR tapos lalabas pg ka scan ung link na ilalagay mo ung image na gusto sa ME-QR kse image lng pwede.
  18. S

    A.i. Wall Art Generator (Free to Generate)

    FYI Studio Maganda to pag mahilig kayo sa mga Wall art, meron din sila Canvas, Tshirt design, Tattoo Design, and Room Design. Sample sa Wall art nila. Pero may bayad if gusto mo ng e download yung design, 1$ per download more or less 55php ito yung gawa ko na...
  19. M

    Help Generate drawing AI

    May mga websites, apps, or similar platforms po ba na gaya ng ChatGPT na kaya mag generate ng art or drawing mismo base sa example statements? Pa comment na lang po, thanks!!!
  20. V

    Help May alam po kayung site na pweding mag generate ng clicks sa iyong link mga paps?

    May alam po kayung site na pweding mag generate ng clicks sa iyong link mga paps? O traffic ba tawag dun?
  21. C

    Help Generate NGiNX Location regex

    Generate NGiNX location regex that matches all in /client except /client/sub
  22. V

    Teach your Chatgpt how to generate your wanted airport code

    ❤️Halimbawa to ng teaching chatgpt by example: Text: “I want to fly from Nashville to Kansas City.” Airport codes: DAL, IDA Text: “I want to fly from Orlando to Boston” Airport codes: Sasagot yan ng accurate at natututo sa by example
  23. K

    Help Best website to generate email replies?

    Meron kayang free website na nag-gegenerate ng email using keywords?
  24. J

    Smart TNT Sun Auto generate imperva

    big credits kay boss Tunneler🙏 Hidden content
  25. J

    Smart TNT Sun Imperva auto generate part2.

    big credits kay boss tunneler . note: respeto ako sakanta labyu. master😘 Hidden content
  26. P

    Help Help Admin Access Globe Wifi B315s-938

    Mga boss baka naman pwedi nyo kong matulongan mag pa admin access ng globe home broadband b315s-938 namin ang hina po kasi ng signal dito sa probinsya baka matulongan nyo ko:cry: Device name: B315s-938 IMEI: 864436035682658 IMSI : 515025575641790 Hardware version: WL2B310M01...
  27. V

    Help Mga paps baka may alam kayung site na makapag generate ng click's sa link?

    Good evening sa inyu mga paps.. Mga paps baka may alam kayung site na makapag generate ng click's sa link? Yung free lng sana may mga bayad kasi mga nakita kong mga site.😅 Thank you in advance po♥️
  28. 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...
  29. S

    Closed 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 !!
  30. 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
  31. 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!
  32. 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>