Return to Answer

Your example from Witten makes one point: generating functions can make differential operators summarize combinatoric/algebraic information -- basically by making differential operators express recurrence relations on the series of coefficients. And generating functions often have nice closed forms, as for example the long known expression of the power generating function for Fibonacci numbers as $x/(1-x-x^2)$. A closed form captures the series as a whole. Closed forms are at least concise information. They can reveal the effect of differential operators, or algebraic relations. Wilf's free book Generatingfunctionology http://www.math.upenn.edu/~wilf/DownldGF.html gives many examples of all this. Because you can use power generating functions, exponential generating functions, Dirichlet series, and more, you have some choice in tailoring the series to meet the application.

Treating generating functions as functions in the usual sense can be useful but is not always possible since the series are not always convergent. As noted Jeff Harvey's reply to Does Physics need non-analytic smooth functions? an important kind of power generating function in physics can have radius of convergence 0.

I would be surprised (and delighted) to see a concise yet comprehensive explanation of all the reasons generating functions work so well. For a brilliant, concise, avowedly not comprehensive effort at this see the preface of Wilf's book. He gives exceptionally clear exposition and motivation throughout that book.

Your example from Witten makes one point: generating functions can make differential operators summarize combinatoric/algebraic information -- basically by making differential operators express recurrence relations on the series of coefficients. And generating functions often have nice closed forms, as for example the long known expression of the power generating function for Fibonacci numbers as $x/(1-x-x^2)$. A closed form captures the series as a whole. Closed forms are at least concise information. They can reveal the effect of differential operators, or algebraic relations. Wilf's free book Generatingfunctionology http://www.math.upenn.edu/~wilf/DownldGF.html gives many examples of all this. Because you can use power generating functions, exponential generating functions, Dirichlet series, and more, you have some choice in tailoring the series to meet the application.

Treating generating functions as functions in the usual sense can be useful but is not always possible since the series are not always convergent. As noted Jeff Harvey's reply to Does Physics need non-analytic smooth functions? an important kind of power generating function in physics can have radius of convergence 0.

I would be surprised (and delighted) to see a concise yet comprehensive explanation of all the reasons generating functions work so well. For a brilliant, concise, avowedly not comprehensive effort at this see the preface of Wilf's book. He gives exceptionally clear exposition and motivation throughout that book.

Your example from Witten makes one point: generating functions can make differential operators summarize combinatoric/algebraic information -- basically by making differential operators express recurrence relations on the series of coefficients. And generating functions often have nice closed forms, as for example the long known expression of the power generating function for Fibonacci numbers as $x/(1-x-x^2)$. A closed form captures the series as a whole. Closed forms are at least concise information. They can reveal the effect of differential operators, or algebraic relations. Wilf's free book Generatingfunctionology http://www.math.upenn.edu/~wilf/DownldGF.html gives many examples of all this. Because you can use ordinary generating functions, exponential generating functions, Dirichlet series, and more, you have some choice in tailoring the series to meet the application.

I would be surprised (and delighted) to see a concise yet comprehensive explanation of all the reasons generating functions work so well. For a brilliant, concise, avowedly not comprehensive effort at this see the preface of Wilf's book. He gives exceptionally clear exposition and motivation throughout that book.

Your example from Witten makes one point: generating functions can make differential operators summarize combinatoric/algebraic information -- basically by making differential operators express recurrence relations on the series of coefficients. And generating functions often have nice closed forms, as for example the long known expression of the power generating function for Fibonacci numbers as $x/(1-x-x^2)$. A closed form captures the series as a whole. Closed forms are at least concise information. They can reveal the effect of differential operators, or algebraic relations. Wilf's free book Generatingfunctionology http://www.math.upenn.edu/~wilf/DownldGF.html gives many examples of all this. Because you can use power generating functions, exponential generating functions, Dirichlet series, and more, you have some choice in tailoring the series to meet the application.

Treating generating functions as functions in the usual sense can be useful but is not always possible since the series are not always convergent. As noted Jeff Harvey's reply to Does Physics need non-analytic smooth functions? an important kind of power generating function in physics can have radius of convergence 0.

I would be surprised (and delighted) to see a concise yet comprehensive explanation of all the reasons generating functions work so well. For a brilliant, concise, avowedly not comprehensive effort at this see the preface of Wilf's book. He gives exceptionally clear exposition and motivation throughout that book.

Your example from Witten makes one point: generating functions can make differential operators summarize combinatoric/algebraic information -- basically by making differential operators express recurrence relations on the series of coefficients. And generating functions often have nice closed forms, as for example the long known expression of the power generating function for Fibonacci numbers as $x/(1-x-x^2)$. A closed form captures the series as a whole. Closed forms are at least concise information. They can reveal the effect of differential operators, or algebraic relations. Wilf's free book Generatingfunctionology http://www.math.upenn.edu/~wilf/DownldGF.html gives many examples of all this. Because you can use ordinary generating functions, exponential generating functions, Dirichlet series, and more, you have some choice in tailoring the series to meet the application.

But I would be surprised (and delighted) to see a concise yet comprehensive explanation of all the reasons generating functions work so well.

Your example from Witten makes one point: generating functions can make differential operators summarize combinatoric/algebraic information -- basically by making differential operators express recurrence relations on the series of coefficients. And generating functions often have nice closed forms, as for example the long known expression of the power generating function for Fibonacci numbers as $x/(1-x-x^2)$. A closed form captures the series as a whole. Closed forms are at least concise information. They can reveal the effect of differential operators, or algebraic relations. Wilf's free book Generatingfunctionology http://www.math.upenn.edu/~wilf/DownldGF.html gives many examples of all this. Because you can use ordinary generating functions, exponential generating functions, Dirichlet series, and more, you have some choice in tailoring the series to meet the application.

I would be surprised (and delighted) to see a concise yet comprehensive explanation of all the reasons generating functions work so well. For a brilliant, concise, avowedly not comprehensive effort at this see the preface of Wilf's book. He gives exceptionally clear exposition and motivation throughout that book.