What is Generating function: Definition and 127 Discussions

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.

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  1. P

    Differentiation of functional integral (Blundell Quantum field theory)

    I am reading the Lancaster & Blundell, Quantum field theory for gifted amateur, p.225 and stuck at understanding some derivations. We will calculate a generating functional for the free scalar field. The free Lagrangian is given by $$ \mathcal{L}_0 = \frac{1}{2}(\partial _\mu \phi)^2 -...
  2. R

    Expanding potential in Legendre polynomials (or spherical harmonics)

    Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid...
  3. S

    B Value of t for Probability Generating Function

    My questions: 1) What about if t = 2? Is there a certain meaning to ##G_X (2)## ? 2) PGF for uniform distribution is ##G_X (t)=\frac{t(1-t^n)}{n(1-t)}## and for t = 1 ##G_X (1)## is undefined so ##G_X (1) =1## is not true for all cases? Thanks
  4. U

    Probability generating function

    (a) I find the geometric distribution $$X~G0(3/8)$$ and I find p to be 0.375 since the mean 0.6 = p/q. So p.g.f of X is $$(5/8)/(1-(3s/8))$$. (b) Not sure how to find the p.g.f of Y once we know there are 6 customers?
  5. chwala

    Probability generating function when x is even

    Homework Statement [/B] A random variable x has a probability function ##G(t)##. Show that the probability that ##x## takes an even value is ## \frac 1 2 ( 1+G(-1))##Homework EquationsThe Attempt at a Solution [/B] ##G(t)= \sum_{k=0}^\infty p_k t^k ##... ## 1=P(X=even)+ P(X=odd)##...1 ##G(-1)=...
  6. D

    Hamilton Jacobi equation for time dependent potential

    Homework Statement Suppose the potential in a problem of one degree of freedom is linearly dependent upon time such that $$H = \frac{p^2}{2m} - mAtx $$ where A is a constant. Solve the dynamical problem by means of Hamilton's principal function under the initial conditions t = 0, x = 0, ##p =...
  7. D

    Problem finding the distribution of holes in a semiconductor

    Homework Statement Long and thin sample of silicon is stationary illuminated with an intensive optical source which can be described by a generation function ##G(x)=\sum_{m=-\infty}^\infty Kδ(x-ma)## (Dirac comb function). Setting is room temperature and ##L_p## and ##D_p## are given. Find the...
  8. Vicol

    A Canonical transformation - derviation problem

    Let me show you part of a book "Mechanics From Newton’s Laws to Deterministic Chaos" by Florian Scheck. I do not understand why these integrands can differ by more than time derivative of some function M. Why doesn't it change the value of integrals? It seems this point is crucial for me to...
  9. P

    Generating Function for Lagrangian Invariant System

    Homework Statement Given a system with a Lagrangian ##L(q,\dot{q})## and Hamiltonian ##H=H(q,p)## and that the Lagrangian is invariant under the transformation ##q \rightarrow q+ K(q) ## find the generating function, G. Homework EquationsThe Attempt at a Solution ##\delta q = \{ q,G \} =...
  10. Sarina3003

    Generating functions, binomial coefficients

    Homework Statement a) I have to find and expression for sequence of $b_n$ in terms of generating functions of the sequence of $a_n$ $$b_n = (-1)^{n}(n+1)a_0 +(-1)^{n-1}n a_1+...+(-1)2a_{n-1}+a_n$$ with $$a_n = a_{n-1} +8a_{n-2} -12a_{n-3} +25(-3)^{n-2} + 32n^2 -64$$ b) I have to use the...
  11. L

    A Bessel function, Generating function

    Generating function for Bessel function is defined by G(x,t)=e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n Why here we have Laurent series, even in case when functions are of real variables?
  12. T

    I Proving a multivariate normal distribution by the moment generating function

    I have proved (8.1). However I am trying to prove that ##\bar{X},X_i-\bar{X},i=1,...,n## has a joint distribution that is multivariate normal. I am trying to prove it by looking at the moment generating function: ##E(e^{t(X_i-\bar{X})}=E(e^{tX_i})E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})## I am...
  13. J

    MHB Finding the Moment Generating Function

    I'm working this problem for my math stat class. Here is what I have for it. First of all, is this the correct method for finding MGF? I thought it was but I don't understand the answers I am getting. How do I determine my values for t? For both I have t not equal to 0 because t is in the...
  14. thecourtholio

    Using generating function to normalize wave function

    Homework Statement Prove that ##\psi_n## in Eq. 2.85 is properly normalized by substituting generating functions in place of the Hermite polynomials that appear in the normalization integral, then equating the resulting Taylor series that you obtain on the two sides of your equation. As a...
  15. dykuma

    Legendre Polynomials & the Generating function

    Homework Statement Homework Equations and in chapter 1 I believe that wanted me to note that The Attempt at a Solution For the first part of this question, as a general statement, I know that P[2 n + 1](0) = 0 will be true as 2n+1 is an odd number, meaning that L is odd, and so the Legendre...
  16. alyafey22

    MHB Generating function for trigamma^2

    In this thread we are looking at the following generating function $$\sum_{n=1}^\infty [\psi_1(n)]^2 y^n$$ We know that this is as hard as evaluating $$\sum_{n=1}^\infty [H_n^{(2)}]^2 y^n$$ This is not a tutorial as I have no idea how to solve for a general formula. I'll keep posting my...
  17. Mark Brewer

    Bessel Generating Function

    Homework Statement Show that the Bessel functions Jn(x) (where n is an integer) have a very nice generating function, namely, G(x,t) := ∑ from -∞ to ∞ of tn Jn(x) = exp {(x/2)((t-T1/t))}, Hint. Starting from the recurrence relation Jn+1(x) + Jn-1(x) = (2n/x)Jn(x), show that G(x,t)...
  18. J

    Same Moment Generating Function, Same Prob. Distribution

    How do you know that if two random variables have the same moment generating function then they have the same probability distribution.
  19. T

    Generating function model

    Homework Statement given one each of u types of candy, two each of v types of candy, and three each of of w types of candy, find a generating function for the number of ways to select r candies. The Attempt at a Solution I am not sure if I understand this correctly, but this is what I came...
  20. Coffee_

    Confirm my reasoning on a generating function proof

    It's about equation (6.5) I'm not entirely getting the reasoning explained by the author so I came up with the following, can anyone confirm or refute. One way to look at equation (6.5) would be: We create variations on the ##q## variables, in the form of ##\delta q(t)##. Since ##Q=Q(q,p,t)##...
  21. Coffee_

    Generating function and Lagrangian invariance

    To make my explanation easier open the ''Generating function approach'' section on this wiki article: http://en.wikipedia.org/wiki/Canonical_transformation The function ##\frac{dG}{dt}## represents the function that always can be added to the Lagrangian without changing the mechanical...
  22. S

    Generating function for the zeta function of the Hamiltonian

    Given a Hamiltonian ##H##, with a spectrum of eigenvalues ##\lambda##, you can define its zeta function as ##\zeta_H(s) = tr \frac{1}{H^s} = \sum_{\lambda}^{} \frac{1}{\lambda^s}##. Subsequently, the log determinant of ##H## with a spectral parameter ##m^2## acts as a generating function for...
  23. throneoo

    Geometric distribution Problem

    Homework Statement a man draws balls from an infinitely large box containing either white and black balls , assume statistical independence. the man draws 1 ball each time and stops once he has at least 1 ball of each color . if the probability of drawing a white ball is p , and and q=1-p is...
  24. M

    Finite Difference Expressed As a Probability Generating Function

    $$F(z) = \sum_{n=0}^\infty a_n x^n $$ $$\partial_zF(z) = \sum_{n=0}^\infty (n+1)a_{n+1}x^n $$ So, we can begin to piece together some differential equations in terms of generating functions in order to satisfy some discrete recursion relation (which is the desired problem to solve). However I...
  25. T

    Probability/Moment Generating Function

    Homework Statement Let X ~ Normal(μ,σ2). Define Y=eX. a) Find the PDF of Y. b) Show that the moment generating function of Y doesn't exist. Homework EquationsThe Attempt at a Solution For part a, I used the fact that fy(y) = |d/dy g-1(y)| fx(g-1(y)). Therefore I got that fy(y)=...
  26. Mogarrr

    Moment generating function DNE

    Homework Statement Write the integral that would define the mgf of the pdf, f(x) = \frac 1{\pi} \frac 1{1+x^2} dx Homework Equations The moment generating function (mgf) is E e^{tX}[\itex]. The Attempt at a Solution My question really has to do with improper integrals. I must...
  27. H

    MHB Generating Function Tree

    Hi, Please I need you help to solve this problem: ---------- Consider a planar tree with $n$ non-root vertices (root edge selected). 1. Give a generating function for vertices distance $d$ from the root. 2. Proof that the total number is $$\displaystyle...
  28. stripes

    Finding the joint moment generating function given the joint PDF

    Homework Statement Homework Equations The Attempt at a Solution The problem is the integral is non-elementary, so now what? Part (b) follows trivially from part (a). But is there some kind of shortcut I have to take, because no matter what substitution I do, the integral...
  29. S

    Difference between generating function and Rodrigue's formula?

    What is the difference between generating function and Rodrigue's formula? Some says that from generating function you can generate required polynomial (say for example from generating function of Legendre polynomial you can get Legendre polynomial.), but in that case,as far as i know, Rodrigues...
  30. A

    Exponential distribution moment generating function to find the mean

    With mean = 2 with exponential distribution Calculate E(200 + 5Y^2 + 4Y^3) = 432 E(200) = 200 E(5Y^2) = 5E(Y^2) = 5(8) = 40 E(4Y^3) = 4E(Y^3) = 4(48) = 192 E(Y^2) = V(Y) + [E(Y)]^2 = 2^2+2^2= 8 E(Y^3) = m_Y^3(0) = 48(1-2(0))^{-4} = 48 is this right?
  31. A

    Puzzle with moment generating function for gamma function

    I am given that The kth moment of an exponential random variable with mean mu is E[Y^k] = k!*mu^k for nonnegative integer k. I found m^2 (0) = (-a)(-a-1)(-beta)^2. The answer I found is however mu^2+a*beta^2 which is different from the k! From the given formula. Could someone help me figure it...
  32. alyafey22

    MHB Generating function of bessel function

    Prove the generating function e^{\frac{x}{2}\left(z-z^{-1}\right)}=\sum_{n=-\infty}^{\infty}J_n(x)z^n
  33. I

    Moment generating function, CDF and density of a random variable

    Assume X is a random variable under a probability space in which the sample space ?= {a,b,c,d,e}. Then if I am told that: X({a}) = 1 X({b}) = 2 X({c}) = 3 X({d}) = 4 X({e}) = 5 And that: P({a}) = P({c}) = P({e}) = 1/10 P({b}) = P({d}) = 7/20 Find the C.D.F of X, the density of X...
  34. K

    What is the moment generating function from a density of a continuous

    Hi everyone, So I am taking a statistics course and finding this concept kinda challenging. wondering if someone can help me with the following problem! Let X be a random variable with probability density function $$f(x)=\begin{cases}xe^{-x} \quad \text{if } x>0\\0 \quad \text{ }...
  35. darida

    Generating Function in Physics

    Could anybody give me some examples of generating function in physics, it's application, and it's use? Thank you
  36. A

    Moment generating function

    Estimation of x i.e. E(x) = Ʃx.p(x) ... p(x) is probabiltiy of x Now my book defines another function mgf(x) i.e. moment generating function of x which is defined as: - mgf(x) = E(etx) I don't understand why was this function defined. Basically we included etx in our function because then...
  37. O

    MHB Moment generating function question

    Let X1,X2,…,Xn be independent random variables that all have the same distribution, let N be an independent non-negative integer valued random variable, and let SN:=X1+X2+⋯+XN. Find an expression for the moment generating function of SN so all i know is that it is i.i.d but i am not sure what...
  38. O

    MHB Generating Function for Gambler's Probs of Broke at Time n

    Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability p and loses one dollar with probability 1 - p. Let fn be the probability that he or she fi rst becomes broke at time n for n = 0, 1, 2... Find a generating function for these...
  39. F

    Using a factorial moment generating function to find probability func.

    Homework Statement Hi everyone! Me and my colleague are working our way through Harold J Larson's "Introduction to Probability Theory and Statistical Inference: Third Edition", and we found something interesting. We both have the same edition of the text, but mine is slightly newer?, and...
  40. O

    MHB Finding expected value from the moment generating function

    Suppose I have the MGF moment generating function mx(t) = (e^t -1)/t How can I find EX?
  41. T

    Moment generating function

    Question A moment-generating function of X is given by M(t) = 0.3e^t + 0.4e^(2t) + 0.2e^(3t) + 0.1e^(5t) Find the pmf of X My attempt x f(x) 1 0.3 2 0.4 3 0.2 4 0 5 0.1 I am just wondering whether it is correct to say "0" for 4 or is it more correct to say x f(x) 1 0.3 2 0.4 3 0.2 5 0.1 or...
  42. M

    Bessel's Function by generating function

    I'm trying to define Bessel's function by using the generating function, I know i need to go through a recursion formula to get there. $e^{(\frac{x}{2}(t-\frac{1}{t})}=\displaystyle\sum_{n=-\infty}^{\infty}J_n(x)t^n$ if this or anyone has latex that's the generating function. Any...
  43. FeDeX_LaTeX

    Moment Generating Function

    Homework Statement Let X be a random variable with a Laplace distribution, so that its probability density function is given by f(x) = \frac{1}{2}e^{-|x|} Sketch f(x). Show that its moment generating function MX (θ) is given by M_{X}(\theta) = \frac{1}{1 - \theta^2} and hence find...
  44. P

    Solve Problem w/ Generating Function e_m(x1-xn)

    I am suppose to use the generating function for e_{m}(x_{1} . . . . x_{n}) to solve a problem. I have tried looking for it but I can not seem to find any information on it. Does anyone know what it is?
  45. A

    Moment Generating Function - Integration Help

    I am working on a probabilty theory problem: Let (X,Y) be distributed with joint density f(x,y)=(1/4)(1+xy(x^2-y^2)) if abs(x)≤1, abs(y)≤1; 0 otherwise Find the MGF of (X,Y). Are X,Y independent? If not, find covariance. I have set up the integral to find the mgf ∫∫e^(sx+ty)f(x,y)dx dy with...
  46. A

    Moment Generating Function Given pdf

    Homework Statement X is distributed exponentially with λa=2. Y is distributed exponentially with λb = 3. X and Y are independent. Let W=max(X,Y), the time until both persons catch their first fish. Let k be a positive integer. Find E(W^k). Also, find P{(1/3)<X/(X+Y)<(1/2)}...
  47. P

    Moment generating function

    given m(t) = (1-p+p*e^t)^5 what is probability P(x<1.23) i know that m(t) = e^tx * f(x) m'(0) = E(X) and m''(0) , can find the var(x) should i calculate it using a normal table?
  48. N

    Probability generating function.

    For any integer valued RV X Summation n=0 to infinity of s^n P(X=<n) = (1-s)^-1 * Summation k=0 to infinity of P(x=k)s^k Why does Sum k=0 to infinity P(x=k)s^k = sum n=0 to infinity of P(X=< n)
  49. S

    Moment generating function to calculate the mean and variance

    I attached a pdf. The questions are not really what is stumping me. Its the wording of the question I don't understand. When it says, "Assume that 0 < λ < 1. Note that your answers will be in terms of the constant λ." and "Assume that λ > 0. Note that your answers will be in terms of the...
  50. P

    Find OGF for Recurrence: a_n = 6a_{n-1} + a_{n-2}, a_0=2, a_1=1

    Find the OGF for the recurrence a_{n}= 6 * a_{n-1}+ a_{n-2} a_{0}=2, a_{1}=1 So here is what I did I said let A = \sum_{2>=n} a_{n}x^{n}then I got A = 6x (A+x) + x^{2}(A +x+2) which gets me A= \frac{6x^2+x^{3} +2x}{1-6x - x^2} ButI should get \frac{2-x}{1-6x - x^2}Can anyone tell me...
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