Bernoulli Numbers – Generating Functions

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In the last of this follow-on series about the Bernoulli Numbers I introduce the Exponential Generating Function for

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Ordinary Generating Functions

One useful way of representing any sequence of numbers (infinite or otherwise) is by setting them as the coefficients of a power series. So for the sequence \( \{a_k\} \) its generating function would be

$$ G(a_k;x)=\sum_{i=0}^{k}a_ix^i=a_0+a_1x+a_2x^2+… $$

As this form is as difficult, if not impossible, to write out in full as the original sequence there not much to be gained unless it can be written is closed form, i.e. a simple function equation.

A quick example to get us going – recall the Catalan numbers which appear in combinatorics and also in my series on the Drunkard’s Walk begin

$$ 1, 1, 2, 5, 14, 42, 132, … $$

Then its OGF (writing \( G(a_k;x)=C(x) \)) would be

$$ C(x)=\sum_{i=0}^{k}c_ix^i=1+x+2x^2+5x^3+14x^4+… $$

It can be shown that its self-convolution

$$ C^2(x)=\sum_{i=0}^{k}c_ix^i=1+2x+5x^2+14x^3+… $$

so that \( C(x)=1+xC^2(x) \).

and solving the resulting quadratic equation, choosing the -ve alternative, gives

$$ C(x)=\frac {\displaystyle 1 – \sqrt{1-4x}}{\displaystyle 2x} $$

This is all very well but how can one recover the infinite polynomial again? Easy some might say, just write the closed version as its Maclaurin series. Easier said than done as one needs L’Hôpital’s rule to cope with the indeterminate forms that regularly crop up. What we need is a Maclaurin series calculator and, of course, one is available online. Here is the result (reading backwards)

The Bernoulli numbers E.G.F.

What about a generating function for the Bernoulli Numbers? Exponential Generating Functions are defined slightly differently but are appropriate to consider

$$ EG(a_k;x)=\sum_{i=0}^{k}a_i\frac{x^i}{i!}=a_0+a_1x+a_2\frac{x^2}{2!}+… $$

So taking the Bernoulli numbers \( \small {B_k} \) with \( B_1=+\frac{ 1}{2} \) and letting

$$ B(x)=\sum_{i=0}^{k}B_i\frac{x^i}{i!}=B_0+B_1x+B_2\frac{x^2}{2!}+B_3\frac{x^3}{3!}+… $$

then considering the Maclaurin series expansion of the exponential function

$$ e^{-x}=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+… $$

and taking the convolution (product of their series expansions) of \( e^{-x} \) with \( B(x) \)

\( e^{-x}B(x)= \)

$$ (1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+…)(B_0+B_1x+B_2\frac{x^2}{2!}+B_3\frac{x^3}{3!}+…) $$

$$ =\sum_{i=0}^{\infty} \left( \sum_{j=0}^{i} {{i} \choose {j}} B_j \right)\frac{\displaystyle x^i}{\displaystyle i!} $$

with \( B_0=1 \) then

\( e^{-x}B(x)= \)

$$ 1-(1-B_1)x+(1-2B_1+B_2)\frac{x^2}{2!}-(1-3B_1+3B_2-B_3)\frac{x^2}{2!}+… $$

and as \( \sum_{i=1}^{n}(-1)^i {{n+1} \choose {i} }B_i=\sum_{i=1}^{n-1}(-1)^i {{n} \choose {i}} B_i+B_n=B_n \)

$$ =-x+(1+B_1x+B_2\frac{x^2}{2!}+B_3\frac{x^3}{3!}+…)=-x+B(x) $$

so

$$ B^{+}(x)=\frac{\displaystyle x.e^x}{\displaystyle e^x-1}=\frac{\displaystyle x}{\displaystyle 1-e^{-x}} $$

Generating its Maclaurin expansion to 12 terms using the Maclaurin series calculator gives (backwards)

which is

$$ =1+B_1x+B_2\frac{x^2}{2!}+B_3\frac{x^3}{3!}+…+B_12\frac{x^12}{12!} $$

Obviously \( 12!(2730)=1307674368000 \) neatly produces the \( B_{12}=-\frac{691}{2730} \) term.

P.S. the EGF of the Bernoulli numbers with \( B_1=-\frac{1}{2} \) is not dissimilar

$$ B^{-}(x)=\frac{\displaystyle x}{\displaystyle e^x-1} $$

I thank Barry Freeman for his blog “Fermat’s Last Theorem” (something I am also very interested in) and 2 posts therein “Bernoulli Numbers“and “A Generating Function for the Bernoulli Numbers” in which he develops the EGF for \( B^{-}(x) \) which I stumbled upon only today (4 Jun, 2024)! I will leave Fermat’s Last Theorem to Barry.

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The SoP series …

Sums of Integer Powers

• Sums of Integers
• Archimedes – Sums of Squares
• Sums of Squares
• Sums of Cubes
• A Sums of Powers Triangle
• Sums of Higher Powers
• Faulberger’s Sums of Powers
• Bernoulli’s Sums of Powers

The Bernoulli Numbers

• The Bernoulli Numbers
• Stirling Numbers
• Worpitzky’s Numbers
• The Bernoulli Numbers redefined
• Correspondence – Luschny & Knuth
• Bernoulli Numbers – Generating Functions

Riemann’s Zeta Function

• Euler & Sums of Reciprocals
• Riemann’s Zeta Function
• The Riemann hypothesis
• Calculating Zeta Zeros