Umbral compositional inversion is a type of duality related to multiplicative and compositional inversion of functions and to matrix inversion, which is very useful in deriving algebraic relations and other identities among polynomial sequences important in number theory, special functions, and enumerative combinatorics as well as operator calculi.

If a pair $p_n(x)$ and $q_n(x)$ of polynomial Sheffer sequences is an umbral inverse pair (UIP), then

$$p_n(q.(x))= x^n=q_n(p.(x)),$$

where, e.g., $p_n(q.(x))=\sum^n_{k=0} \; p_{n,k} \cdot q_k(x).$

This implies that the pair of lower triangular matrices comprised of the coefficients of these polynomial sequences are an inverse pair.

An important UIP of binomial Sheffer sequences are the Bell / Touchard polynomials, comprised of the Stirling numbers of the second kind, and the falling factorials, comprised of the Stirling numbers of the first kind. An important UIP of Appell Sheffer sequences are the Bernoulli polynomials and the reciprocal integer polynomials.

For a binomial Sheffer sequence of polynomials, defined by the exponential generating function

$$e^{h(t)x}=e^{p.(x)t},$$

with $h(0)=0$ and $(p.(x)^n)=p_n(x)$, the umbral compositional inverse Sheffer sequence is given by

$$e^{h^{(-1)}(t)x}=e^{q.(x)t},$$

where $h$ and $h^{(-1)}$ are a compositional inverse pair. E.g., $(h(t),h^{-1}(t))=(e^t-1,\ln(1+t))$ generate the Bell and falling factorial pair.

For an Appell Sheffer sequence, defined by the e.g.f.

$$f(t)e^{x \cdot t}= e^{u.(x)t},$$

with $f(0)=1$, its umbral compositional inverse Appell sequence is given by

$$\frac{1}{f(t)}e^{x \cdot t} = e^{v.(x)t}.$$

For example, $f(t)=\frac{t}{e^t-1}$ for the Bernoulli polynomials $B_n(x)$, so

$$e^{v.(x)t}=\frac{e^t-1}{t}e^{xt}$$ for the UI dual, the reciprocal integer polynomials,

$$v_n(x)=\frac{(1+x)^{n+1}-x^{n+1}}{n+1}.$$

Consequently, using the simple convolution properties of Appell sequences and the UI relation,

$$ \frac{(1+B.(x))^{n+1}-B.(x)^{n+1}}{n+1} = \frac{(B.(1+x))^{n+1}-B.(x)^{n+1}}{n+1} = x^n = \frac{d}{dx} \frac{x^{n+1}}{n+1},$$

so we can formally identify for any power series

$$S(B.(1+x))-S(B.(x)) = S^{'}(x).$$

There are many interesting and useful dual operators associated with these UIPs (e.g., the first derivative and the forward finite difference operator and its inverse are related to the Bernoulli UIP), and the umbral inverse property alone is often very useful in giving simple, concise derivations of properties of the polynomials and their coefficients. Interweaving the two types of inversions, multiplicative and compositional, a simple formula for the Bernoulli polynomials and their associated base numbers, the Bernoulli numbers, in terms of the Stirling numbers is easily derived in my blog post Compositional Inverse Operators and Sheffer Sequences.

Umbral compositional inversion is a type of duality related to multiplicative and compositional inversion of functions and to matrix inversion, which is very useful in deriving algebraic relations and other identities among polynomial sequences important in number theory, special functions, and enumerative combinatorics as well as operator calculi.

If a pair $p_n(x)$ and $q_n(x)$ of polynomial Sheffer sequences is an umbral inverse pair (UIP), then

$$p_n(q.(x))= x^n=q_n(p.(x)),$$

where, e.g., $p_n(q.(x))=\sum^n_{k=0} \; p_{n,k} \cdot q_k(x).$

This implies that the pair of lower triangular matrices comprised of the coefficients of these polynomial sequences are an inverse pair.

An important UIP of binomial Sheffer sequences are the Bell / Touchard polynomials, comprised of the Stirling numbers of the second kind, and the falling factorials, comprised of the Stirling numbers of the first kind. An important UIP of Appell Sheffer sequences are the Bernoulli polynomials and the reciprocal integer polynomials.

For a binomial Sheffer sequence of polynomials, defined by the exponential generating function

$$e^{h(t)x}=e^{p.(x)t},$$

with $h(0)=0$ and $(p.(x)^n)=p_n(x)$, the umbral compositional inverse Sheffer sequence is given by

$$e^{h^{(-1)}(t)x}=e^{q.(x)t},$$

where $h$ and $h^{(-1)}$ are a compositional inverse pair. E.g., $(h(t),h^{-1}(t))=(e^t-1,\ln(1+t))$ generate the Bell and falling factorial pair.

For an Appell Sheffer sequence, defined by the e.g.f.

$$f(t)e^{x \cdot t}= e^{u.(x)t},$$

with $f(0)=1$, its umbral compositional inverse Appell sequence is given by

$$\frac{1}{f(t)}e^{x \cdot t} = e^{v.(x)t}.$$

For example, $f(t)=\frac{t}{e^t-1}$ for the Bernoulli polynomials $B_n(x)$, so

$$e^{v.(x)t}=\frac{e^t-1}{t}e^{xt}$$ for the UI dual, the reciprocal integer polynomials,

$$v_n(x)=\frac{(1+x)^{n+1}-x^{n+1}}{n+1}.$$

Consequently, using the simple convolution properties of Appell sequences and the UI relation,

$$ \frac{(1+B.(x))^{n+1}-B.(x)^{n+1}}{n+1} = \frac{(B.(1+x))^{n+1}-B.(x)^{n+1}}{n+1} = x^n = \frac{d}{dx} \frac{x^{n+1}}{n+1},$$

so we can formally identify for any power series

$$S(B.(1+x))-S(B.(x)) = S^{'}(x).$$

There are many interesting and useful dual operators associated with these UIPs (e.g., the first derivative and the forward finite difference operator and its inverse are related to the Bernoulli UIP), and the umbral inverse property alone is often very useful in giving simple, concise derivations of properties of the polynomials and their coefficients. Interweaving the two types of inversions, multiplicative and compositional, a simple formula for the Bernoulli polynomials and their associated base numbers, the Bernoulli numbers, in terms of the Stirling numbers is easily derived in my blog post Compositional Inverse Operators and Sheffer Sequences.

Umbral compositional inversion is a type of duality related to multiplicative and compositional inversion of functions and to matrix inversion, which is very useful in deriving identities and algebraic relations among polynomial sequences important in number theory, special functions, and enumerative combinatorics as well as operator calculi.

If a pair $p_n(x)$ and $q_n(x)$ of polynomial Sheffer sequences is an umbral inverse pair (UIP), then

$$p_n(q.(x))= x^n=q_n(p.(x)),$$

where, e.g., $p_n(q.(x))=\sum^n_{k=0} \; p_{n,k} \cdot q_k(x).$

This implies that the pair of lower triangular matrices comprised of the coefficients of these polynomial sequences are an inverse pair.

An important UIP of binomial Sheffer sequences are the Bell / Touchard polynomials, comprised of the Stirling numbers of the second kind, and the falling factorials, comprised of the Stirling numbers of the first kind. An important UIP of Appell Sheffer sequences are the Bernoulli polynomials and the reciprocal integer polynomials.

For a binomial Sheffer sequence of polynomials, defined by the exponential generating function

$$e^{h(t)x}=e^{p.(x)t},$$

with $h(0)=0$ and $(p.(x)^n)=p_n(x)$, the umbral compositional inverse Sheffer sequence is given by

$$e^{h^{(-1)}(t)x}=e^{q.(x)t},$$

where $h$ and $h^{(-1)}$ are a compositional inverse pair. E.g., $(h(t),h^{-1}(t))=(e^t-1,\ln(1+t))$ generate the Bell and falling factorial pair.

For an Appell Sheffer sequence, defined by the e.g.f.

$$f(t)e^{x \cdot t}= e^{u.(x)t},$$

with $f(0)=1$, its umbral compositional inverse Appell sequence is given by

$$\frac{1}{f(t)}e^{x \cdot t} = e^{v.(x)t}.$$

For example, $f(t)=\frac{t}{e^t-1}$ for the Bernoulli polynomials, so

$$e^{v.(x)t}=\frac{e^t-1}{t}e^{xt}$$ for the UI dual, the reciprocal integer polynomials $v_n(x)=\frac{(1+x)^{n+1}-x^{n+1}}{n+1}$.

There are many interesting and useful dual operators associated with these UIPs (e.g., the first derivative and the forward finite difference operator and its inverse are related to the Bernoulli UIP), and the umbral inverse property alone is often very useful in giving simple, concise derivations of properties of the polynomials and their coefficients. Interweaving the two types of inversions, multiplicative and compositional, a simple formula for the Bernoulli polynomials and their associated base numbers, the Bernoulli numbers, in terms of the Stirling numbers is easily derived in my blog post Compositional Inverse Operators and Sheffer Sequences.

Umbral compositional inversion is a type of duality related to multiplicative and compositional inversion of functions and to matrix inversion, which is very useful in deriving algebraic relations and other identities among polynomial sequences important in number theory, special functions, and enumerative combinatorics as well as operator calculi.

If a pair $p_n(x)$ and $q_n(x)$ of polynomial Sheffer sequences is an umbral inverse pair (UIP), then

$$p_n(q.(x))= x^n=q_n(p.(x)),$$

where, e.g., $p_n(q.(x))=\sum^n_{k=0} \; p_{n,k} \cdot q_k(x).$

This implies that the pair of lower triangular matrices comprised of the coefficients of these polynomial sequences are an inverse pair.

An important UIP of binomial Sheffer sequences are the Bell / Touchard polynomials, comprised of the Stirling numbers of the second kind, and the falling factorials, comprised of the Stirling numbers of the first kind. An important UIP of Appell Sheffer sequences are the Bernoulli polynomials and the reciprocal integer polynomials.

For a binomial Sheffer sequence of polynomials, defined by the exponential generating function

$$e^{h(t)x}=e^{p.(x)t},$$

with $h(0)=0$ and $(p.(x)^n)=p_n(x)$, the umbral compositional inverse Sheffer sequence is given by

$$e^{h^{(-1)}(t)x}=e^{q.(x)t},$$

where $h$ and $h^{(-1)}$ are a compositional inverse pair. E.g., $(h(t),h^{-1}(t))=(e^t-1,\ln(1+t))$ generate the Bell and falling factorial pair.

For an Appell Sheffer sequence, defined by the e.g.f.

$$f(t)e^{x \cdot t}= e^{u.(x)t},$$

with $f(0)=1$, its umbral compositional inverse Appell sequence is given by

$$\frac{1}{f(t)}e^{x \cdot t} = e^{v.(x)t}.$$

For example, $f(t)=\frac{t}{e^t-1}$ for the Bernoulli polynomials $B_n(x)$, so

$$e^{v.(x)t}=\frac{e^t-1}{t}e^{xt}$$ for the UI dual, the reciprocal integer polynomials,

$$v_n(x)=\frac{(1+x)^{n+1}-x^{n+1}}{n+1}.$$

Consequently, using the simple convolution properties of Appell sequences and the UI relation,

$$ \frac{(1+B.(x))^{n+1}-B.(x)^{n+1}}{n+1} = \frac{(B.(1+x))^{n+1}-B.(x)^{n+1}}{n+1} = x^n = \frac{d}{dx} \frac{x^{n+1}}{n+1},$$

so we can formally identify for any power series

$$S(B.(1+x))-S(B.(x)) = S^{'}(x).$$

There are many interesting and useful dual operators associated with these UIPs (e.g., the first derivative and the forward finite difference operator and its inverse are related to the Bernoulli UIP), and the umbral inverse property alone is often very useful in giving simple, concise derivations of properties of the polynomials and their coefficients. Interweaving the two types of inversions, multiplicative and compositional, a simple formula for the Bernoulli polynomials and their associated base numbers, the Bernoulli numbers, in terms of the Stirling numbers is easily derived in my blog post Compositional Inverse Operators and Sheffer Sequences.