# 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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3. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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?
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### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

35. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...