Bell series

In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell.

Given an arithmetic function ${\displaystyle f}$ and a prime ${\displaystyle p}$, define the formal power series ${\displaystyle f_{p}(x)}$, called the Bell series of ${\displaystyle f}$ modulo ${\displaystyle p}$ as:

${\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}.}$

Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem: given multiplicative functions ${\displaystyle f}$ and ${\displaystyle g}$, one has ${\displaystyle f=g}$ if and only if:

${\displaystyle f_{p}(x)=g_{p}(x)}$ for all primes ${\displaystyle p}$.

Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions ${\displaystyle f}$ and ${\displaystyle g}$, let ${\displaystyle h=f*g}$ be their Dirichlet convolution. Then for every prime ${\displaystyle p}$, one has:

${\displaystyle h_{p}(x)=f_{p}(x)g_{p}(x).\,}$

In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.

If ${\displaystyle f}$ is completely multiplicative, then formally:

${\displaystyle f_{p}(x)={\frac {1}{1-f(p)x}}.}$

Examples

The following is a table of the Bell series of well-known arithmetic functions.

• The Möbius function ${\displaystyle \mu }$ has ${\displaystyle \mu _{p}(x)=1-x.}$
• The Mobius function squared has ${\displaystyle \mu _{p}^{2}(x)=1+x.}$
• Euler's totient ${\displaystyle \varphi }$ has ${\displaystyle \varphi _{p}(x)={\frac {1-x}{1-px}}.}$
• The multiplicative identity of the Dirichlet convolution ${\displaystyle \delta }$ has ${\displaystyle \delta _{p}(x)=1.}$
• The Liouville function ${\displaystyle \lambda }$ has ${\displaystyle \lambda _{p}(x)={\frac {1}{1+x}}.}$
• The power function Idk has ${\displaystyle ({\textrm {Id}}_{k})_{p}(x)={\frac {1}{1-p^{k}x}}.}$ Here, Idk is the completely multiplicative function ${\displaystyle \operatorname {Id} _{k}(n)=n^{k}}$.
• The divisor function ${\displaystyle \sigma _{k}}$ has ${\displaystyle (\sigma _{k})_{p}(x)={\frac {1}{1-(1+p^{k})x+p^{k}x^{2}}}.}$
• The unit function satisfies ${\displaystyle 1_{p}(x)=(1-x)^{-1}}$, i.e., is the geometric series.
• If ${\displaystyle f(n)=2^{\omega (n)}=\sum _{d|n}\mu ^{2}(d)}$ is the power of the prime omega function, then ${\displaystyle f_{p}(x)={\frac {1+x}{1-x}}.}$
• Suppose that f is multiplicative and g is any arithmetic function satisfying ${\displaystyle f(p^{n+1})=f(p)f(p^{n})-g(p)f(p^{n-1})}$ for all primes p and ${\displaystyle n\geq 1}$. Then ${\displaystyle f_{p}(x)=\left(1-f(p)x+g(p)x^{2}\right)^{-1}.}$
• If ${\displaystyle \mu _{k}(n)=\sum _{d^{k}|n}\mu _{k-1}\left({\frac {n}{d^{k}}}\right)\mu _{k-1}\left({\frac {n}{d}}\right)}$ denotes the Mobius function of order k, then ${\displaystyle (\mu _{k})_{p}(x)={\frac {1-2x^{k}+x^{k+1}}{1-x}}.}$

References

• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
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Generating function

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series. This formal power series is the generating function. Unlike an ordinary series, this formal series is allowed to diverge, meaning that the generating function is not always a true function and the "variable" is actually 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 series in more than one indeterminate, to encode information about arrays of numbers indexed by several natural 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.

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