Bell series explained

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.

f and a prime p , define the formal power series f _{p(x)} , called the Bell series of f modulo p as:f _{p(x)=\sum} ^{inf ty} ^{n =0} 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

f and g , one has f = g if and only if:f _{p(x)=g} _{p(x)} for all primes p .

Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions

f and g , let h = f*g be their Dirichlet convolution. Then for every prime p , one has: 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

f is completely multiplicative, then formally: f _{p(x) = (1)/(1- f(p)x)} .

Examples

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

\mu has \mu _{p(x)=1-x.}

• The Mobius function squared has

\mu ^{2 (x)} _{p} = 1+ x. \varphi has \varphi _{p(x) = (1- x)/(1- px)} . \delta has \delta _{p(x)=1.} λ has λ _{p(x) = (1)/(1+ x)} .

• The power function Id_k has

( rm{Id} _{k)} ^{p(x) = (1)/(1- p ^{kx})} . Here, Id_k is the completely multiplicative function \operatorname{Id} ^{k} _{k(n) = n} . \sigma _{k} has (\sigma _{k)} ^{p(x) = (1)/(( 1- p ^{kx)(1-x)})} .

• The constant function, with value 1, satisfies

1 _{p(x)} = ( 1- x) ^{-1} , i.e., is the geometric series.

• If

f(n) = 2 ^{\omega(n)} = \sum _{d|n} \mu ^{2(d)} is the power of the prime omega function, then f _{p(x)} = (1+ x)/(1- x) .

• Suppose that f is multiplicative and g is any arithmetic function satisfying

f(p ^{n+1} ) = f(p) f(p ^{n)} - g(p) f(p ^{n-1} ) for all primes p and n \geq 1 . Then f _{p(x)} = \left( 1- f(p)x + g(p)x ^{2\right)} ^{-1} .

• If

\mu _{k(n)} = \sum _{d ^{k|n}} \mu _{k-1} \left( (n)/(d ^{k}) \right) \mu _{k-1} \left( (n)/(d) \right) denotes the Möbius function of order k, then (\mu _{k)} _{p(x)} = (1-2 x ^{k+x} ^{k+1})/(1- x) .

See also

• Bell numbers