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. C

    Calculating the mean and variance from a moment generating function

    Homework Statement Assume that X is squared-Chi-distributed, which means that the moment generating function is given by: m(t)=(1-2t)^{-k/2} Use the mgf to find E(X) and var(X) The Attempt at a Solution I know that m'(0)=E(X), and m''(0)=var(X). So I find...
  2. J

    Cannot tell when a probability generating function converges for |s|<1

    Hi, I have a problem that is already solved... I thought 3 of the 4 functions were probability generating functions, but I got one wrong and don't know why. The solution says g(s)=1+s-s^2 is not a probability generating function. However, g(1)=1 and I think g(s) converges to 1 for |s|<1...
  3. N

    Generating function for groups of order n

    I've done some searching and have thus far come up empty handed, so I'm hoping that someone here knows something that I don't. I'm wondering if there has been any work on the enumeration of groups of order n (up to isomorphism); specifically, has anyone derived a generating function? Ideally...
  4. P

    MHB Probability generating function

    My question is : can a pgf have a constant term? The reason I ask is that I was asked to show the (time) derivative of a pgf was equal to some multiple of the pgf and hence show the pgf was as given. So naturally , I differentiated the given answer and showed it satisfied the equation. But...
  5. P

    What is the generating function proof for Legendre polynomials?

    Hey I've been trying to show that \frac{1}{\sqrt{1+u^2 -2xu}} is a generating function of the polynomials, in other words that \frac{1}{\sqrt{1+u^2 -2xu}}=\sum\limits_{n=0}^{\infty }{{{P}_{n}}(x){{u}^{n}}} My class was told to do this by first finding the binomial series of...
  6. P

    Legendre poly, generating function

    hey guys, my lecturer skipped the proof to show that \frac{1}{\sqrt{1+u^2 -2xu}} is a generating function of the polynomials, he told us that we should do it as an exercise by first finding the binomial series of \frac{1}{\sqrt{1-s}} then insert s = -u2 + 2xu he then said to expand...
  7. D

    Generating function, hamiltonian dynamics

    Homework Statement A canonical transformation is made from (p,q) to (P,Q) through a generating function F=a*cot(Q), where 'a' is a constant. Express p,q in terms of P,Q. Homework Equations The Attempt at a Solution A generating function is supposed to be a bridge between (p,q) and...
  8. L

    A problem while verifying the generating function of Legendre Polynomials.

    Our professor gave us an a problem to solve, she asked us to prove or verify the following identity: http://img818.imageshack.us/img818/5082/6254.png Where \Phi is the Generating function of Legendre polynomials given by: \Phi(x,h)= (1 - 2hx + h2)-1/2 2. This Identity is from...
  9. G

    Generating function of n-point function

    Hey guys, I have a doubt. I was wondering if it is possible to have a generating function Z[J] where its integral has not a linear dependence on J(t), but a quadratic or even cubic dependence, like Z[J]=∫Dq exp{S[q] + ∫ J²(t) q(t)dt}, and how this would alter the calculation of the n-point...
  10. G

    N-point function calculation from a generating function

    If a have ln Z[J] = ∫ J²f(t)dt+ a∫ J³dt + b∫ J^4dt, where J=J(t), and I would like to get the 3-point and 4-point functions, how do I proceed? I have tried to use the regular formula for the n-point function, when you derive Z[J] n times in relation to J(t_1)...J(t_n) and after applies J=0...
  11. fluidistic

    Hamiltonian, generating function, canonical transformation

    Homework Statement Consider a harmonic oscillator with generalized coordinates q and p with a frequency omega and mass m. Let the transformation (p,q) -> (Q,P) be such that F_2(q,P,t)=\frac{qP}{\cos \theta }-\frac{m\omega }{2}(q^2+P^2)\tan \theta. 1)Find K(Q,P) where \theta is a function of...
  12. fluidistic

    Finding a generating function for a canonical transformation

    Homework Statement I'm trying to find a generating function for the canonical transformation Q=\left ( \frac{\sin p}{q} \right ), P=q \cot p.Homework Equations I am not really sure. I know there are 4 different types of generating function. I guess it's totally up to me to choose the type of...
  13. fluidistic

    Canonical transformations, generating function

    Homework Statement Given the generating function F=\sum _i f_i (q_j,t)P_i, 1)Find the corresponding canonical transformations. 2)Show that the transformations of generalized coordinates are canonical transformations. 3)What meaning does the canonical transformation originated by the generating...
  14. W

    Method of Moment Generating Function Help

    Let X1 be a binomial random variable with n1 trials and p1 = 0.2 and X2 be an independent binomial random variable with n2 trials and p2 = 0.8. Find the probability function of Y = X1 + n2 – X2. Exactly how does one calculate the mgf of (n2 - X2)?
  15. W

    Joint Moment Generating Function Help

    Hi, I've no idea where to go with the question below: Joint moment generating function of X and Y - MXY(s,t) = 1/(1-2s-3t+6st) for s<1/2, t<1/3. Find P(min(X,Y) > 0.95) and P(max(X,Y) > 0.8)
  16. S

    Probability Generating Function / Geometric

    Homework Statement a) P(X=x)=pq^x,\,x\geq 0 Find the PGF. b) P(X=x)=pq^{|x|},\,x\,\epsilon\,\text{Z} Find the PGF. 2. The attempt at a solution a) G_X(s)=E(s^X)=\displaystyle\sum_{x\geq 0}pq^x s^x=p\displaystyle\sum_{x\geq 0}(qs)^x=\frac{p}{1-qs} b) Not sure about this one... Is it: as...
  17. J

    Moment Generating Function of normally distributed variable

    Hi guys, I need to find the moment generating function for X ~ N (0,1) and then also the MGF for X2 . I know how to do the first part but I'm unsure for X2. do i use the identity that if Y = aX then MY(t) = E(eY(t)) = E(e(t)aX) or do i just square 2pi-1/2e x2/2 and then solve as...
  18. E

    Generating Function: Formula for Nth Term?

    Does exist general formula for nth term in sequence if I have generative function? In my case, generative function is (1/(1-x))-(1/(1-x^3)).
  19. E

    Find the nth Term of a Generating Function

    If I know generating function of a series, what formula gives nth term? Specifically, my generating function is f(x)=(Ʃ(k=1, to m-1) x^k)/(1-x^m) ***The function represent series: 0,1,1,...,1,0,1,1,...,1,0,... where m is period; i.e. 0,1,1,0,1,1,0 m=3***
  20. E

    Generating Function for 0,1 Sequences with Period m

    What is generating function of these sequences: 0,1,0,1...or 0,1,1,0,1,1,0...or 0,1,1,1,0,1,1,1,0... where m is period, in first example m=2; in the second m=3, and so on. Generating function must be general, so I can just put m.
  21. T

    Probability generating function for random variable

    Homework Statement A random variable X has probability generating function gX(s) = (5-4s2)-1 Calculate P(X=3) and P(X=4) Homework Equations The Attempt at a Solution Ehh don't really know where to go with one... I know: gX(s) = E(sx) = Ʃ p(X=k)(sk) Nit sure how to proceed.. Any help would...
  22. T

    Adding two distributions with same moment generating function

    Homework Statement I wanted to know what the result would be if you added two distributions with the same moment generating function. For example, what would the result be of: Mx(t) + My(t) if Mx(t) = (1/3 + 2/3et) and My(t) = (1/3 + 2/3et) Homework Equations The Attempt at a...
  23. S

    Exploring the Exponential Distribution: Generating Function Method

    Homework Statement F_{X}(x)= λe^{-λx} \;for\; x>0 \;\;\;and \;0 \;otherwise After finding the characteristic function for the Exponential Distribution, which is (I could do this without problem); F_{X}(k)=λ(λ-ik)^{-1} Now the question is; Let X_1,X_2,\ldots,X_i be i.i.d. exponential...
  24. T

    Best way to integrate a moment generating function?

    Homework Statement ∫etxx2e-x Homework Equations M(t) = etx f(x) dx The Attempt at a Solution I know the solution is -1/(t-3)3, however I'm having difficulty integrating the function. UV - ∫ V DU is extremely long and challenging, I'm wondering if there is a shortcut (i.e. quotient rule?)...
  25. T

    Probability Mass Function/Moment Generating Function

    Homework Statement The pmf of a random variable X is given by f(x) = π(1 − π)x for x = 0, 1, ..., ∞, and 0 ≤ π ≤ 1. a) Show that this function actually is a pmf. b) Find E(X). c) Find the moment generating function of X, MX(t) = E(etX). 2. The attempt at a solution My solution was done...
  26. J

    Generating function of a recurrance relation

    Suppose A(x) is a generating function for the sequence a0, a1, a2, . . . that satisfies the recurrence a[n+2] = −a[n+1] + 6a[n] for n > 0, with initial conditions a[0] = 2 and a[1] = −1. Find a formula for A(x) and use it to find an explicit formula for a[n]. I don't know what I am doing...
  27. W

    Moment generating function problem

    From the pdf of X, f(x) = 1/8 e^-x/8, x > 0, find the mgf of Y=X/4 +1. What is then the value of P(2.3 < Y < 4.1)? Homework Statement Homework Equations Moment generating function of exponential distributionThe Attempt at a Solution I have the mgf of X, which is 1/8 / (1/8 - t). I have also...
  28. A

    Derivation of bessel generating function

    Homework Statement The bessel generating function: exp(x*(t-(1/t))/2)=sum from 0 to n(Jn(x)t^(n)) Homework Equations The Attempt at a Solution exp(x*(t-(1/t))/2)=exp((x/2)*t)exp((x/2)*(1/t)) used the McLaurin expansion of exponentials. Not sure how to bring the powers equal to that...
  29. B

    Calculating PDF from MGF: Advice Needed

    My goal here is to at least approximately calculate the probability density function (PDF) given the moment generating function (MGF), M_X(t). I have managed to calculate the exact form of the MGF as an infinite series in t. In principle, if I replace t with it and perform an inverse...
  30. M

    Derive Hermite Polynomial Generating Function from Recurrence Relation

    I am not sure how to format in LaTeX; I apologize for that. The Hermite polynomials Hn(x) (physicist's version) satisfy the recurrence relation, H_{n+1}(x) - 2xHn(x) + 2nH_{n-1}(x) = 0; H0(x) = 1 and H1(x) = 2x: Use this to derive the generating function for the Hermite polynomials...
  31. T

    Hamilton's Equations and Generating Function

    Say we have a Hamiltonian H(q,p,t) and we then transform from p and q to P=P(q,p,t) and Q=Q(q,p,t), with: P\dot{Q}-K=p\dot{q}-H+\frac{d}{dt}F(q,p,Q,P,t) where K is the new Hamiltonian. How do we show that P and Q obey Hamilton's equations with Hamiltonian K? I have tried partial...
  32. Oxymoron

    Moment Generating Function (proof of definition)

    Homework Statement Prove that for a random variable X with continuous probability distribution function f_X(x) that the Moment Generating Function, defined as M_X(t) := E[e^{tX}] is M_X(t) = \int_x^{\infty}e^{tx}f_X(x)dx Homework Equations Above and E[X] =...
  33. S

    Combinatorics: Finding a Generating Function

    Homework Statement A national singing contest has 5 distinct entrants from each state. Use a generating function for modeling the number of ways to pick 20 semifinalists if: a) There is at most 1 person from each state b) There are at most 3 people from each state. Homework...
  34. T

    What is the mistake in this attempt to prove \sum {\frac{x^n}{n}} = -ln(1-x)?

    Homework Statement I am trying to show that \sum{ \frac{x^n}{n}} = -ln(1-x) But I am doing something wrong and I can't find my mistake. Please find my mistake and let me know what it is. Thanks The Attempt at a Solution set f(x)=\sum {\frac{x^n}{n}} then f'(x)= \sum {x^n-1} so...
  35. S

    Probability generating function

    Homework Statement Let Y=x+4. Compute rY(t) in terms of rX Homework Equations The Attempt at a Solution is the answer just r 3X+4 (t) ?
  36. M

    Finding the Probability distribution function given Moment Generating Function

    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! Suppose X is a discrete random variable with moment generating function M(t) = 2/10 + 1/10e^t + 2/10e^(2t) + 3/10e^(3t) + 2/10e^(4t)...
  37. D

    Definition of moment generating function

    M(t)=E(e^{ty})=\sum_{y=0}^{n} e^{ty}p(y) Is this correct?
  38. M

    Moment generating function help?

    I know that hte MGF is = the E[e^tx] How do i show that if i take a sample (X1;X2; : : :Xn) from the exponential density f(x) = A*e^(-Ax), then the sum Z = sum(Xi) has the gamma density? I found that the MGF for the exponential was A/(t-A) if that helps Thanks
  39. N

    Generating function for Legendre polynomials

    Homework Statement Using binomial expansion, prove that \frac{1}{\sqrt{1 - 2 x u + u^2}} = \sum_{k} P_k(x) u^k. Homework Equations \frac{1}{\sqrt{1 + v}} = \sum_{k} (-1)^k \frac{(2k)!}{2^{2k} (k!)^2} v^k The Attempt at a Solution I simply inserted v = u^2 - 2 x u, then...
  40. D

    Is the Cumulant Generating Function correctly defined?

    Cumulative generating function is K(t)=K_1(t)t+K_2(t)\frac{t^2}{2!}+K_3(t)\frac{t^3}{3!}+... where K_{n}(t)=K^{(n)}(t) Now K(t)=ln M(t)=ln E(e^{ty})=ln E(f(0)+f'(0)\frac {t}{1!}+f''(0)\frac{t^2}{2!}+...)=ln E(1+\frac{t}{1!}y+\frac{t^2}{2!} y^2+...)=ln [1+\frac{t}{1!} E(Y)+\frac{t^2}{2!}...
  41. D

    Is This Moment Generating Function Expression Correct?

    Is the following correct? M(t)=1+t\mu'_1+\frac{t^2}{2!}\mu'_2+\frac{t^3}{3!}\mu'_3+... =\sum_{n=0}^{\infty} \frac {E(Y^n)t^n}{n!} where \mu'_n=E(Y^n)
  42. E

    Moment generating function of a continuous variable

    Homework Statement Find the moment generating of: f(x)=.15e^{-.15x} Homework Equations M_x(t)= \int_{-\infty}^{\infty}{e^{tx}f(x)dx} The Attempt at a Solution I get down to the point (if I've done my calculus correctly) and gotten: \frac{.15e^{(t-.15)x}}{t-.15} \Bigr|...
  43. J

    Finding expected value from moment generating function

    Find E(X) given the moment generating function M_X (t) = 1 / (1-t^2) for |t| < 1. (The pdf is f(x) = 0.5*exp(-|x|), for all x, so graphically you can see that E(X) should be 0.) ---- I know that E(X) = M ' _X (t) = 0 BUT M ' _X (t) = 2x / (1-x^2)^2 which is indeterminate at 0...
  44. Z

    Can You Derive the MGF of Y=-X Using MGF of X Directly?

    Say r.v. X, we have pdf f(x) and mgf Mx(t) defined. Then define Y=-X, y is negative x. Can we get mgf of Y, i.e. My(t) and how? I know I can go the way to get pdf f(y) first then My(t). I want to know if Mx(t) is already in my hands, it should be easier to get My(t) other than do f(y)...
  45. L

    What is a generating function (GF) in physics?

    I am trying to understand CM wrt QFT and found out that I need to understand the HJE. This brought me to reading about all related subjects. The history lesson alone has been awesome. However, now I am reading about the HJE and found the Wikipedia pages lacking as to exactly what is the...
  46. Z

    Moment generating function (Probability)

    given a probability distribution P(x) >0 on a given interval , if we define the moment generatign function M(x)= \int_{a}^{b}dt e^{xt}P(t)dt my question is , if the moment problem is determined, then could we say that ALL the zeros of M(x) are PURELY imaginary ? ia or this is only for...
  47. L

    Generating function for canonical transformation

    Homework Statement Given the transformation Q = p+iaq, P = \frac{p-iaq}{2ia} Homework Equations find the generating function The Attempt at a Solution As far as I know, one needs to find two independent variables and try to solve. I couldn't find such to variables. I've...
  48. E

    Probability generating function

    Homework Statement A random variable X has the generating function f(z) = 1 / (2-z)^2 Find E(X) and Var(X). Homework Equations The Attempt at a Solution Would anyone explain in simpler terms the notion of the generating function, such that I may be able to solve...
  49. S

    Probability generating function (binomial distribution)

    Homework Statement The probabilty generating funtion G is definied for random varibles whos range are \subset {0,1,2,3,...}. If Y is such a random variable we will call it a counting random varible. Its probabiltiy generating function is G(s) = E(s^{y}) for those s's such that E(|s|^{y})) <...
  50. D

    Generating Function- change for a dollar

    Homework Statement If our currency consists of a two-cent coin and three kinds of pennies, how many ways can we make change for a dollar? Homework Equations The Attempt at a Solution the previous part to this problem led to this generating function...
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