Generating function transformation#Fractional integrals and derivatives

{{main|Series multisection}}

Series multisection provides formulas for generating functions enumerating the sequence $\backslash \{f\_\{an+b\}\backslash \}$ given an ordinary generating function $F(z)$ where $a,\; b\; \backslash in\; \backslash mathbb\{N\}$, $a\; \backslash geq\; 2$, and . In the first two cases where $(a,\; b)\; :=\; (2,\; 0),\; (2,\; 1)$, we can expand these arithmetic progression generating functions directly in terms of $F(z)$:

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; f\_\{2n\}\; z^\{2n\}\; =\; \backslash frac\{1\}\{2\}\backslash left(F(z)\; +\; F(-z)\backslash right)$

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; f\_\{2n+1\}\; z^\{2n+1\}\; =\; \backslash frac\{1\}\{2\}\backslash left(F(z)\; -\; F(-z)\backslash right).$

More generally, suppose that $a\; \backslash geq\; 3$ and that $\backslash omega\_a\; :=\; \backslash exp\backslash left(\backslash frac\{2\backslash pi\backslash imath\}\{a\}\backslash right)$ denotes the $a^\{th\}$ primitive root of unity. Then we have the following formula,See Section 1.2.9 in Knuth's The Art of Computer Programming (Vol. 1). often known as the root of unity filter:

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; f\_\{an+b\}\; z^\{an+b\}\; =\; \backslash frac\{1\}\{a\}\; \backslash times\; \backslash sum\_\{m=0\}^\{a-1\}\; \backslash omega\_a^\{-mb\}\; F\backslash left(\backslash omega\_a^\{m\}z\backslash right).$

For integers $m\; \backslash geq\; 1$, another useful formula providing somewhat reversed floored arithmetic progressions are generated by the identitySolution to exercise 7.36 on page 569 in Graham, Knuth and Patshnik.

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; f\_\{\backslash lfloor\; \backslash frac\{n\}\{m\}\; \backslash rfloor\}\; z^n\; =\; \backslash frac\{1-z^m\}\{1-z\}\; F(z^m)\; =\; \backslash left(1+z+\backslash cdots+z^\{m-2\}\; +\; z^\{m-1\}\backslash right)\; F(z^m).$

The exponential Bell polynomials, $B\_\{n,k\}(x\_1,\; \backslash ldots,\; x\_n)\; :=\; n!\; \backslash cdot\; [t^n\; u^k]\; \backslash Phi(t,\; u)$, are defined by the exponential generating functionSee section 3.3 in Comtet.

:$\backslash Phi(t,\; u)\; =\; \backslash exp\backslash left(u\; \backslash times\; \backslash sum\_\{m\; \backslash geq\; 1\}\; x\_m\; \backslash frac\{t^m\}\{m!\}\backslash right)\; =$

1 + \sum_{n \geq 1} \left\{\sum_{k=1}^{n} B_{n,k}(x_1, x_2, \ldots) u^k\right\} \frac{t^n}{n!}.

The next formulas for powers, logarithms, and compositions of formal power series are expanded by these polynomials with variables in the coefficients of the original generating functions.See sections 3.3–3.4 in Comtet.See section 1.9(vi) in the NIST Handbook. The formula for the exponential of a generating function is given implicitly through the Bell polynomials by the EGF for these polynomials defined in the previous formula for some sequence of $\backslash \{x\_i\backslash \}$.

The power series for the reciprocal of a generating function, $F(z)$, is expanded by

:$\backslash frac\{1\}\{F(z)\}\; =\; \backslash frac\{1\}\{f\_0\}\; -\; \backslash frac\{f\_1\}\{f\_0^2\}\; z\; +\; \backslash frac\{\backslash left(f\_1^2-f\_0\; f\_2\backslash right)\}\{f\_0^3\}\; z^2\; -$

\frac{f_1^3-2f_0f_1f_2+f_0^2f_3}{f_0^4} + \cdots.

If we let $b\_n\; :=\; [z^n]\; 1\; /\; F(z)$ denote the coefficients in the expansion of the reciprocal generating function, then we have the following recurrence relation:

:$b\_n\; =\; -\; \backslash frac\{1\}\{f\_0\}\backslash left(f\_1\; b\_\{n-1\}\; +\; f\_2\; b\_\{n-2\}\; +\; \backslash cdots\; +\; f\_n\; b\_0\backslash right),\; n\; \backslash geq\; 1.$

Let $m\; \backslash in\; \backslash mathbb\{C\}$ be fixed, suppose that $f\_0\; =\; 1$, and denote $b\_n^\{(m)\}\; :=\; [z^n]\; F(z)^m$. Then we have a series expansion for $F(z)^m$ given by

:$F(z)^m\; =\; 1\; +\; m\; f\_1\; z\; +\; m\backslash left((m-1)f\_1^2+2f\_2\backslash right)\; \backslash frac\{z^2\}\{2\}\; +$

\left(m(m-1)(m-2)f_1^3 +6m(m-1) f_2+6mf_3\right) \frac{z^3}{6} + \cdots,

and the coefficients $b\_n^\{(m)\}$ satisfy a recurrence relation of the form

:$n\; \backslash cdot\; b\_n^\{(m)\}\; =\; (m-n+1)\; f\_1\; b\_\{n-1\}^\{(m)\}\; +\; (2m-n+2)\; f\_2\; b\_\{n-2\}^\{(m)\}\; +\; \backslash cdots\; +\; ((n-1)m-1)\; f\_\{n-1\}\; b\_1^\{(m)\}\; +\; n\; m\; f\_n,\; n\; \backslash geq\; 1.$

Another formula for the coefficients, $b\_n^\{(m)\}$, is expanded by the Bell polynomials as

:$F(z)^m\; =\; f\_0^m\; +\; \backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash left(\backslash sum\_\{1\; \backslash leq\; k\; \backslash leq\; n\}\; (m)\_k\; f\_0^\{m-k\}\; B\_\{n,k\}(f\_1\; \backslash cdot\; 1!,\; f\_2\; \backslash cdot\; 2!,\; \backslash ldots)\backslash right)\; \backslash frac\{z^n\}\{n!\},$

where $(r)\_n$ denotes the Pochhammer symbol.

If we let $f\_0\; =\; 1$ and define $q\_n\; :=\; [z^n]\; \backslash log\; F(z)$, then we have a power series expansion for the composite generating function given by

:$\backslash log\; F(z)\; =\; f\_1\; +\; \backslash left(2f\_2-f\_1^2\backslash right)\; \backslash frac\{z\}\{2\}\; +\; \backslash left(3f\_3-3f\_1f\_2+f\_1^3\backslash right)\; \backslash frac\{z^2\}\{3\}\; +\; \backslash cdots,$

where the coefficients, $q\_n$, in the previous expansion satisfy the recurrence relation given by

:$n\; \backslash cdot\; q\_n\; =\; n\; f\_n\; -\; (n-1)f\_1\; q\_\{n-1\}\; -\; (n-2)f\_2\; q\_\{n-2\}\; -\; \backslash cdots\; -\; f\_\{n-1\}\; q\_1,$

and a corresponding formula expanded by the Bell polynomials in the form of the power series coefficients of the following generating function:

:$\backslash log\; F(z)\; =\; \backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash left(\backslash sum\_\{1\; \backslash leq\; k\; \backslash leq\; n\}\; (-1)^\{k-1\}\; (k-1)!\; B\_\{n,k\}(f\_1\; \backslash cdot\; 1!,\; f\_2\; \backslash cdot\; 2!,\; \backslash ldots)\backslash right)\; \backslash frac\{z^n\}\{n!\}.$

{{main|Faà di Bruno's formula}}

Let $\backslash widehat\{F\}(z)$ denote the EGF of the sequence, $\backslash \{f\_n\backslash \}\_\{n\; \backslash geq\; 0\}$, and suppose that $\backslash widehat\{G\}(z)$ is the EGF of the sequence, $\backslash \{g\_n\backslash \}\_\{n\; \backslash geq\; 0\}$. Faà di Bruno's formula implies that the sequence, $\backslash \{h\_n\backslash \}\_\{n\; \backslash geq\; 0\}$, generated by the composition $\backslash widehat\{H\}(z)\; :=\; \backslash widehat\{F\}(\backslash widehat\{G\}(z))$, can be expressed in terms of the exponential Bell polynomials as follows:

:$h\_n\; =\; \backslash sum\_\{1\; \backslash leq\; k\; \backslash leq\; n\}\; f\_k\; \backslash cdot\; B\_\{n,k\}(g\_1,\; g\_2,\; \backslash cdots,\; g\_\{n-k+1\})\; +\; f\_0\; \backslash cdot\; \backslash delta\_\{n,0\}.$

We have the following integral formulas for $a,\; b\; \backslash in\; \backslash mathbb\{Z\}^\{+\}$ which can be applied termwise with respect to $z$ when $z$ is taken to be any formal power series variable:See page 566 of Graham, Knuth and Patashnik for the statement of the last conversion formula.

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; f\_n\; z^n\; =\; \backslash int\_0^\{\backslash infty\}\; \backslash widehat\{F\}(tz)\; e^\{-t\}\; dt\; =\; z^\{-1\}\; \backslash mathcal\{L\}[\backslash widehat\{F\}](z^\{-1\})$

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; \backslash Gamma(an+b)\; \backslash cdot\; f\_n\; z^n\; =\; \backslash int\_0^\{\backslash infty\}\; t^\{b-1\}\; e^\{-t\}\; F(t^a\; z)\; dt.$

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; \backslash frac\{f\_n\}\{n!\}\; z^n\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_\{-\backslash pi\}^\{\backslash pi\}\; F\backslash left(z\; e^\{-\backslash imath\backslash vartheta\}\backslash right)\; e^\{e^\{\backslash imath\backslash vartheta\}\}\; d\backslash vartheta.$

Notice that the first and last of these integral formulas are used to convert between the EGF to the OGF of a sequence, and from the OGF to the EGF of a sequence whenever these integrals are convergent.

The first integral formula corresponds to the Laplace transform (or sometimes the formal Laplace–Borel transformation) of generating functions, denoted by $\backslash mathcal\{L\}[F](z)$, defined in.See Appendix B.13 of Flajolet and Sedgewick. Other integral representations for the gamma function in the second of the previous formulas can of course also be used to construct similar integral transformations. One particular formula results in the case of the double factorial function example given immediately below in this section. The last integral formula is compared to Hankel's loop integral for the reciprocal gamma function applied termwise to the power series for $F(z)$.

The single factorial function, $(2n)!$, is expressed as a product of two double factorial functions of the form

:$(2n)!\; =\; (2n)!!\; \backslash times\; (2n-1)!!\; =\; \backslash frac\{4^n\; \backslash cdot\; n!\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash times\; \backslash Gamma\backslash left(n+\backslash frac\{1\}\{2\}\backslash right),$

where an integral for the double factorial function, or rational gamma function, is given by

:$\backslash frac\{1\}\{2\}\; \backslash cdot\; (2n-1)!!\; =\; \backslash frac\{2^n\}\{\backslash sqrt\{4\backslash pi\}\}\; \backslash Gamma\backslash left(n+\backslash frac\{1\}\{2\}\backslash right)\; =$

\frac{1}{\sqrt{2\pi}} \times \int_0^{\infty} e^{-t^2 / 2} t^{2n} \, dt,

for natural numbers $n\; \backslash geq\; 0$. This integral representation of $(2n-1)!!$ then implies that for fixed non-zero $q\; \backslash in\; \backslash mathbb\{C\}$ and any integral powers $k\; \backslash geq\; 0$, we have the formula

:$\backslash frac\{\backslash log(q)^k\}\{k!\}\; =\; \backslash frac\{1\}\{(2k)!\}\; \backslash times\; \backslash left[\backslash int\_0^\{\backslash infty\}\; \backslash frac\{2\; e^\{-t^2/2\}\}\{\backslash sqrt\{2\backslash pi\}\}$

\left(\sqrt{2\log(q)} \cdot t\right)^{2k} \, dt\right].

Thus for any prescribed integer $j\; \backslash geq\; 0$, we can use the previous integral representation together with the formula for extracting arithmetic progressions from a sequence OGF given above, to formulate the next integral representation for the so-termed modified Stirling number EGF as

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; \backslash left\backslash \{\backslash begin\{matrix\}\; 2n\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\; \backslash frac\{\backslash log(q)^n\}\{n!\}\; =$

\int_0^{\infty} \frac{e^{-t^2/2}}{\sqrt{2\pi} \cdot j!}\left[\sum_{b = \pm 1}

\left(e^{b\sqrt{2 \log(q)} \cdot t}-1\right)^j\right] dt,

which is convergent provided suitable conditions on the parameter .Refer to the proof of Theorem 2.3 in Math.NT/1609.02803.

For fixed non-zero $c,\; z\; \backslash in\; \backslash mathbb\{C\}$ defined such that , let the geometric series over the non-negative integral powers of $(cz)^n$ be denoted by $G(z)\; :=\; 1\; /\; (1-cz)$. The corresponding higher-order $j^\{th\}$ derivatives of the geometric series with respect to $z$ are denoted by the sequence of functions

:$G\_j(z)\; :=\; \backslash frac\{(cz)^j\}\{1-cz\}\; \backslash times\; \backslash left(\backslash frac\{d\}\{dz\}\backslash right)^\{(j)\}\backslash left[G(z)\backslash right],$

for non-negative integers $j\; \backslash geq\; 0$. These $j^\{th\}$ derivatives of the ordinary geometric series can be shown, for example by induction, to satisfy an explicit closed-form formula given by

:$G\_j(z)\; =\; \backslash frac\{(cz)^j\; \backslash cdot\; j!\}\{(1-cz)^\{j+2\}\},$

for any $j\; \backslash geq\; 0$ whenever . As an example of the third OGF $\backslash longmapsto$ EGF conversion formula cited above, we can compute the following corresponding exponential forms of the generating functions $G\_j(z)$:

:$\backslash widehat\{G\}\_j(z)\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_\{-\backslash pi\}^\{+\backslash pi\}\; G\_j\backslash left(z\; e^\{-\backslash imath\; t\}\backslash right)\; e^\{e^\{\backslash imath\; t\}\}\; dt\; =$

\frac{(cz)^j e^{cz}}{(j+1)}\left(j+1+z\right).

Fractional integrals and fractional derivatives (see the main article) form another generalized class of integration and differentiation operations that can be applied to the OGF of a sequence to form the corresponding OGF of a transformed sequence. For $\backslash Re(\backslash alpha)\; >\; 0$ we define the fractional integral operator (of order $\backslash alpha$) by the integral transformationSee section 1.15(vi)–(vii) in the NIST Handbook.

:$I^\{\backslash alpha\}\; F(z)\; =\; \backslash frac\{1\}\{\backslash Gamma(\backslash alpha)\}\; \backslash int\_0^\{z\}\; (z-t)^\{\backslash alpha-1\}\; F(t)\; dt,$

which corresponds to the (formal) power series given by

:$I^\{\backslash alpha\}\; F(z)\; =\; \backslash sum\_\{n\; \backslash geq\; 0\}\; \backslash frac\{n!\}\{\backslash Gamma(n+\backslash alpha+1)\}\; f\_n\; z^\{n+\backslash alpha\}.$

For fixed $\backslash alpha,\; \backslash beta\; \backslash in\; \backslash mathbb\{C\}$ defined such that $\backslash Re(\backslash alpha),\; \backslash Re(\backslash beta)\; >\; 0$, we have that the operators $I^\{\backslash alpha\}\; I^\{\backslash beta\}\; =\; I^\{\backslash alpha+\backslash beta\}$.

Moreover, for fixed $\backslash alpha\; \backslash in\; \backslash mathbb\{C\}$ and integers $n$ satisfying we can define the notion of the fractional derivative satisfying the properties that

:$D^\{\backslash alpha\}\; F(z)\; =\; \backslash frac\{d^\{(n)\}\}\{dz^\{(n)\}\}\; I^\{n-\backslash alpha\}F(z),$

and

:$D^\{k\}\; I^\{\backslash alpha\}\; =\; D^\{n\}\; I^\{\backslash alpha+n-k\}$ for $k\; =\; 1,\; 2,\; \backslash ldots,\; n,$

where we have the semigroup property that $D^\{\backslash alpha\}\; D^\{\backslash beta\}\; =\; D^\{\backslash alpha+\backslash beta\}$ only when none of $\backslash alpha,\; \backslash beta,\; \backslash alpha+\backslash beta$ is integer-valued.

For fixed $s\; \backslash in\; \backslash mathbb\{Z\}^\{+\}$, we have that (compare to the special case of the integral formula for the Nielsen generalized polylogarithm function defined in{{cite web|last1=Weisstein|first1=Eric W.|title=Nielsen Generalized Polylogarithm|url=http://mathworld.wolfram.com/NielsenGeneralizedPolylogarithm.html|website=MathWorld}}) See equation (4) in section 2 of Borwein, Borwein and Girgensohn's article Explicit evaluation of Euler sums (1994).

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; \backslash frac\{f\_n\}\{(n+1)^s\}\; z^n\; =\; \backslash frac\{(-1)^\{s-1\}\}\{(s-1)!\}\; \backslash int\_0^1\; \backslash log^\{s-1\}(t)\; F(tz)\; dt.$

Notice that if we set $g\_n\; \backslash equiv\; f\_\{n+1\}$, the integral with respect to the generating function, $G(z)$, in the last equation when $z\; \backslash equiv\; 1$ corresponds to the Dirichlet generating function, or DGF, $\backslash widetilde\{F\}(s)$, of the sequence of $\backslash \{f\_n\backslash \}$ provided that the integral converges. This class of polylogarithm-related integral transformations is related to the derivative-based zeta series transformations defined in the next sections.

For fixed non-zero $q,\; c,\; z\; \backslash in\; \backslash mathbb\{C\}$ such that and , we have the following integral representations for the so-termed square series generating function associated with the sequence $\backslash \{f\_n\backslash \}$, which can be integrated termwise with respect to $z$:See the article Math.NT/1609.02803.

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; q^\{n^2\}\; f\_n\; \backslash cdot\; (cz)^n\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_0^\{\backslash infty\}\; e^\{-t^2/2\}\backslash left[F\backslash left(e^\{t\backslash sqrt\{2\backslash log(q)\}\}\; \backslash cdot\; cz\backslash right)\; +\; F\backslash left(e^\{-t\; \backslash sqrt\{2\backslash log(q)\}\}\; \backslash cdot\; cz\backslash right)\backslash right]\; dt.$

This result, which is proved in the reference, follows from a variant of the double factorial function transformation integral for the Stirling numbers of the second kind given as an example above. In particular, since

:$q^\{n^2\}\; =\; \backslash exp(n^2\; \backslash cdot\; \backslash log(q))\; =\; 1\; +\; n^2\; \backslash log(q)\; +\; n^4\; \backslash frac\{\backslash log(q)^2\}\{2!\}\; +\; n^6\; \backslash frac\{\backslash log(q)^3\}\{3!\}\; +\; \backslash cdots,$

we can use a variant of the positive-order derivative-based OGF transformations defined in the next sections involving the Stirling numbers of the second kind to obtain an integral formula for the generating function of the sequence, $\backslash left\backslash \{S(2n,\; j)\; /\; n!\backslash right\backslash \}$, and then perform a sum over the $j^\{th\}$ derivatives of the formal OGF, $F(z)$ to obtain the result in the previous equation where the arithmetic progression generating function at hand is denoted by

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; \backslash left\backslash \{\backslash begin\{matrix\}\; 2n\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\; \backslash frac\{z^\{2n\}\}\{(2n)!\}\; =\; \backslash frac\{1\}\{2\; j!\}\backslash left((e^z-1)^j\; +\; (e^\{-z\}-1)^j\backslash right),$

for each fixed $j\; \backslash in\; \backslash mathbb\{N\}$.

We have an integral representation for the Hadamard product of two generating functions, $F(z)$ and $G(z)$, stated in the following form:

:$(F\; \backslash odot\; G)(z)\; :=\; \backslash sum\_\{n\; \backslash geq\; 0\}\; f\_n\; g\_n\; z^n\; =$

\frac{1}{2\pi} \int_0^{2\pi} F\left(\sqrt{z} e^{i t}\right) G\left(\sqrt{z} e^{-i t}\right) dt,

where i is the imaginary unit.

More information about Hadamard products as diagonal generating functions of multivariate sequences and/or generating functions and the classes of generating functions these diagonal OGFs belong to is found in Stanley's book.See section 6.3 in Stanley's book. The reference also provides nested coefficient extraction formulas of the form

:$\backslash operatorname\{diag\}\backslash left(F\_1\; \backslash cdots\; F\_k\backslash right)\; :=\; \backslash sum\_\{n\; \backslash geq\; 0\}\; f\_\{1,n\}\; \backslash cdots\; f\_\{k,n\}\; z^n\; =$

[x_{k-1}^0 \cdots x_2^0 x_1^0] F_k\left(\frac{z}{x_{k-1}}\right)

F_{k-1}\left(\frac{x_{k-1}}{x_{k-2}}\right) \cdots F_2\left(\frac{x_2}{x_1}\right) F_1(x_1),

which are particularly useful in the cases where the component sequence generating functions, $F\_i(z)$, can be expanded in a Laurent series, or fractional series, in $z$, such as in the special case where all of the component generating functions are rational, which leads to an algebraic form of the corresponding diagonal generating function.

In general, the Hadamard product of two rational generating functions is itself rational.See section 2.4 in Lando's book. This is seen by noticing that the coefficients of a rational generating function form quasi-polynomial terms of the form

:$f\_n\; =\; p\_1(n)\; \backslash rho\_1^n\; +\; \backslash cdots\; +\; p\_\{\backslash ell\}(n)\; \backslash rho\_\{\backslash ell\}^n,$

where the reciprocal roots, $\backslash rho\_i\; \backslash in\; \backslash mathbb\{C\}$, are fixed scalars and where $p\_i(n)$ is a polynomial in $n$ for all $1\; \backslash leq\; i\; \backslash leq\; \backslash ell$.

For example, the Hadamard product of the two generating functions

:$F(z)\; :=\; \backslash frac\{1\}\{1+a\_1\; z+a\_2\; z^2\}$

and

:$G(z)\; :=\; \backslash frac\{1\}\{1+b\_1\; z+b\_2\; z^2\}$

is given by the rational generating function formula{{cite journal|first1=E. A. |last1=Potekhina| title=Application of Hadamard product to some combinatorial and probabilistic problems|journal=Discr. Math. Appl.|year=2017|volume=27|number=3|pages=177–186|doi=10.1515/dma-2017-0020|s2cid=125969602}}

:$(F\; \backslash odot\; G)(z)\; =\; \backslash frac\{1-a\_2b\_2\; z^2\}\{1-a\_1b\_1\; z\; +\; \backslash left(a\_2b\_1^2+a\_1^2b\_2-a\_2b\_2\backslash right)\; z^2\; -a\_1a\_2b\_1b\_2\; z^3\; +\; a\_2^2b\_2^2\; z^4\}.$

Ordinary generating functions for generalized factorial functions formed as special cases of the generalized rising factorial product functions, or Pochhammer k-symbol, defined by

:$p\_n(\backslash alpha,\; R)\; :=\; R(R+\backslash alpha)\; \backslash cdots\; (R+(n-1)\backslash alpha)\; =\; \backslash alpha^n\; \backslash cdot\; \backslash left(\backslash frac\{R\}\{\backslash alpha\}\backslash right)\_n,$

where $R$ is fixed, $\backslash alpha\; \backslash neq\; 0$, and $(x)\_n$ denotes the Pochhammer symbol are generated (at least formally) by the Jacobi-type J-fractions (or special forms of continued fractions) established in the reference.{{cite journal|first1=M. D. |last1=Schmidt|title=Jacobi type continued fractions for ordinary generating functions of generalized factorial functions|year=2017|journal = J. Int. Seq. | volume=20|pages=17.3.4|url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html|arxiv=1610.09691}} If we let $\backslash operatorname\{Conv\}\_h(\backslash alpha,\; R;\; z)\; :=\; \backslash operatorname\{FP\}\_h(\backslash alpha,\; R;\; z)\; /\; \backslash operatorname\{FQ\}\_h(\backslash alpha,\; R;\; z)$ denote the $h^\backslash text\{th\}$ convergent to these infinite continued fractions where the component convergent functions are defined for all integers $h\; \backslash geq\; 2$ by

:$\backslash operatorname\{FP\}\_h(\backslash alpha,\; R;\; z)\; =\; \backslash sum\_\{n=0\}^\{h-1\}\backslash left[\backslash sum\_\{k=0\}^\{n\}\; \backslash binom\{h\}\{k\}\; \backslash left(1-h-\backslash frac\{R\}\{\backslash alpha\}\backslash right)\_k\; \backslash left(\backslash frac\{R\}\{\backslash alpha\}\backslash right)\_\{n-k\}\backslash right]\; (\backslash alpha\; z)^n,$

and

:$$

\begin{align}

\operatorname{FQ}_h(\alpha, R; z) & = (-\alpha z)^h \cdot h! \times L_h^{\left(R / \alpha-1\right)}\left((\alpha z)^{-1}\right) \\

& = \sum_{k=0}^{h} \binom{h}{k} \left[\prod_{j=0}^{k-1} (R+(j-1-j)\alpha)\right] (-z)^k,

\end{align}

where $L\_n^\{(\backslash beta)\}(x)$ denotes an associated Laguerre polynomial, then we have that the $h^\{th\}$ convergent function, $\backslash operatorname\{Conv\}\_h(\backslash alpha,\; R;\; z)$, exactly enumerates the product sequences, $p\_n(\backslash alpha,\; R)$, for all . For each $h\; \backslash geq\; 2$, the $h^\{th\}$ convergent function is expanded as a finite sum involving only paired reciprocals of the Laguerre polynomials in the form of

:$\backslash operatorname\{Conv\}\_h(\backslash alpha,\; R;\; z)\; =\; \backslash sum\_\{i=0\}^\{h-1\}\; \backslash binom\{\backslash frac\{R\}\{\backslash alpha\}+i-1\}\{i\}\; \backslash times$

\frac{(-\alpha z)^{-1}}{(i+1) \cdot L_i^{\left(R/\alpha-1\right)}\left((\alpha z)^{-1}\right)

L_{i+1}^{\left(R/\alpha-1\right)}\left((\alpha z)^{-1}\right)}

Moreover, since the single factorial function is given by both $n!\; =\; p\_n(1,\; 1)$ and $n!\; =\; p\_n(-1,\; n)$, we can generate the single factorial function terms using the approximate rational convergent generating functions up to order $2h$. This observation suggests an approach to approximating the exact (formal) Laplace–Borel transform usually given in terms of the integral representation from the previous section by a Hadamard product, or diagonal-coefficient, generating function. In particular, given any OGF $G(z)$ we can form the approximate Laplace transform, which is $2h$-order accurate, by the diagonal coefficient extraction formula stated above given by

:$$

\begin{align}

\widetilde{\mathcal{L}}_h[G](z) & := [x^0] \operatorname{Conv}_h\left(1, 1; \frac{z}{x}\right) G(x) \\

&\ = \frac{1}{2\pi} \int_0^{2\pi}

\operatorname{Conv}_h\left(1, 1; \sqrt{z} e^{I t}\right) G\left(\sqrt{z} e^{-I t}\right) dt.

\end{align}

Examples of sequences enumerated through these diagonal coefficient generating functions arising from the sequence factorial function multiplier provided by the rational convergent functions include

:$$

\begin{align}

n!^2 & = [z^n][x^0] \operatorname{Conv}_h\left(-1, n; \frac{z}{x}\right) \operatorname{Conv}_h\left(-1, n; x\right), h \geq n \\

\binom{2n}{n} & = [x_1^0 x_2^0 z^n] \operatorname{Conv}_h\left(-2, 2n; \frac{z}{x_2}\right)

\operatorname{Conv}_h\left(-2, 2n-1; \frac{x_2}{x_1}\right)

I_0(2\sqrt{x_1}) \\

\binom{3n}{n} \binom{2n}{n} & = [x_1^0 x_2^0 z^n] \operatorname{Conv}_h\left(-3, 3n-1; \frac{3z}{x_2}\right)

\operatorname{Conv}_h\left(-3, 3n-2; \frac{x_2}{x_1}\right)

I_0(2\sqrt{x_1}) \\

!n & = n! \times \sum_{i=0}^{n} \frac{(-1)^i}{i!} =

[z^n x^0] \left(\frac{e^{-x}}{(1-x)} \operatorname{Conv}_n\left(-1, n; \frac{z}{x}\right)\right) \\

\operatorname{af}(n) & = \sum_{k=1}^{n} (-1)^{n-k} k! = [z^n]\left(\frac{\operatorname{Conv}_n(1, 1; z)-1}{1+z}\right) \\

(t-1)^n P_n\left(\frac{t+1}{t-1}\right) & = \sum_{k=0}^{n} \binom{n}{k}^2 t^k \\

& =

[x_1^0 x_2^0] [z^n] \left(\operatorname{Conv}_n\left(1, 1; \frac{z}{x_1}\right)

\operatorname{Conv}_n\left(1, 1; \frac{x_1}{x_2}\right)

I_0(2\sqrt{t \cdot x_2}) I_0(2\sqrt{x_2})\right), n \geq 1 \\

(2n-1)!! & = \sum_{k=1}^{n} \frac{(n-1)!}{(k-1)!} k \cdot (2k-3)!! \\

& = [x_1^0 x_2^0 x_3^{n-1}]\left(\operatorname{Conv}_n\left(1, 1; \frac{x_3}{x_2}\right)

\operatorname{Conv}_n\left(2, 1; \frac{x_2}{x_1}\right)

\frac{(x_1+1) e^{x_1}}{(1-x_2)}\right),

\end{align}

where $I\_0(z)$ denotes a modified Bessel function, $!n$ denotes the subfactorial function, $\backslash operatorname\{af\}(n)$ denotes the alternating factorial function, and $P\_n(x)$ is a Legendre polynomial. Other examples of sequences enumerated through applications of these rational Hadamard product generating functions given in the article include the Barnes G-function, combinatorial sums involving the double factorial function, sums of powers sequences, and sequences of binomials.

For fixed $k\; \backslash in\; \backslash mathbb\{Z\}^\{+\}$, we have that if the sequence OGF $F(z)$ has $j^\{th\}$ derivatives of all required orders for $1\; \backslash leq\; j\; \backslash leq\; k$, that the positive-order zeta series transformation is given bySee the inductive proof given in section 2 of Math.NT/1609.02803.

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; n^k\; f\_n\; z^n\; =\; \backslash sum\_\{j=0\}^\{k\}\; \backslash left\backslash \{\backslash begin\{matrix\}\; k\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\; z^j\; F^\{(j)\}(z),$

where $\backslash scriptstyle\{\backslash left\backslash \{\backslash begin\{matrix\}\; n\; \backslash \backslash \; k\; \backslash end\{matrix\}\; \backslash right\backslash \}\}$ denotes a Stirling number of the second kind.

In particular, we have the following special case identity when $f\_n\; \backslash equiv\; 1\; \backslash forall\; n$ when $\backslash scriptstyle\{\backslash left\backslash langle\backslash begin\{matrix\}\; n\; \backslash \backslash \; m\; \backslash end\{matrix\}\; \backslash right\backslash rangle\}$ denotes the triangle of first-order Eulerian numbers:See the table in section 7.4 of Graham, Knuth and Patashnik.

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; n^k\; z^n\; =\; \backslash sum\_\{j=0\}^\{k\}\; \backslash left\backslash \{\backslash begin\{matrix\}\; k\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\; \backslash frac\{z^j\; \backslash cdot\; j!\}\{(1-z)^\{j+1\}\}\; =$

\frac{1}{(1-z)^{k+1}} \times \sum_{0 \leq m < k} \left\langle\begin{matrix} k \\ m \end{matrix} \right\rangle z^{m+1}.

We can also expand the negative-order zeta series transformations by a similar procedure to the above expansions given in terms of the $j^\{th\}$-order derivatives of some $F(z)\; \backslash in\; C^\{\backslash infty\}$ and an infinite, non-triangular set of generalized Stirling numbers in reverse, or generalized Stirling numbers of the second kind defined within this context.

In particular, for integers $k,\; j\; \backslash geq\; 0$, define these generalized classes of Stirling numbers of the second kind by the formula

:$\backslash left\backslash \{\backslash begin\{matrix\}\; k+2\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\_\{\backslash ast\}\; :=\; \backslash frac\{1\}\{j!\}\; \backslash times\; \backslash sum\_\{m=1\}^\{j\}\; \backslash binom\{j\}\{m\}\; \backslash frac\{(-1)^\{j-m\}\}\{m^\{k\}\}.$

Then for $k\; \backslash in\; \backslash mathbb\{Z\}^\{+\}$ and some prescribed OGF, $F(z)\; \backslash in\; C^\{\backslash infty\}$, i.e., so that the higher-order $j^\{th\}$ derivatives of $F(z)$ exist for all $j\; \backslash geq\; 0$, we have that

:$\backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{f\_n\}\{n^k\}\; z^n\; =\; \backslash sum\_\{j\; \backslash geq\; 1\}\; \backslash left\backslash \{\backslash begin\{matrix\}\; k+2\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\_\{\backslash ast\}\; z^j\; F^\{(j)\}(z).$

A table of the first few zeta series transformation coefficients, $\backslash scriptstyle\{\backslash left\backslash \{\backslash begin\{matrix\}\; k\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\_\{\backslash ast\}\}$, appears below. These weighted-harmonic-number expansions are almost identical to the known formulas for the Stirling numbers of the first kind up to the leading sign on the weighted harmonic number terms in the expansions.

The next series related to the polylogarithm functions (the dilogarithm and trilogarithm functions, respectively), the alternating zeta function and the Riemann zeta function are formulated from the previous negative-order series results found in the references. In particular, when $s\; :=\; 2$ (or equivalently, when $k\; :=\; 4$ in the table above), we have the following special case series for the dilogarithm and corresponding constant value of the alternating zeta function:

:$$

\begin{align}

\text{Li}_2(z) & = \sum_{j \geq 1} \frac{(-1)^{j-1}}{2} \left(H_j^2+H_j^{(2)}\right) \frac{z^j}{(1-z)^{j+1}} \\

\zeta^{\ast}(2) & = \frac{\pi^2}{12} = \sum_{j \geq 1} \frac{\left(H_j^2+H_j^{(2)}\right)}{4 \cdot 2^j}.

\end{align}

When $s\; :=\; 3$ (or when $k\; :=\; 5$ in the notation used in the previous subsection), we similarly obtain special case series for these functions given by

:$$

\begin{align}

\text{Li}_3(z) & = \sum_{j \geq 1} \frac{(-1)^{j-1}}{6} \left(H_j^3+3H_j H_j^{(2)} + 2 H_j^{(3)}\right) \frac{z^j}{(1-z)^{j+1}} \\

\zeta^{\ast}(3) & = \frac{3}{4} \zeta(3) = \sum_{j \geq 1} \frac{\left(H_j^3+3H_j H_j^{(2)} + 2 H_j^{(3)}\right)}{12 \cdot 2^j} \\

& = \frac{1}{6} \log(2)^3 + \sum_{j \geq 0} \frac{H_j H_j^{(2)}}{2^{j+1}}.

\end{align}

It is known that the first-order harmonic numbers have a closed-form exponential generating function expanded in terms of the natural logarithm, the incomplete gamma function, and the exponential integral given by

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; \backslash frac\{H\_n\}\{n!\}\; z^n\; =\; e^z\; \backslash left(\; \backslash mbox\{E\}\_1(z)\; +\; \backslash gamma\; +\; \backslash log\; z\backslash right)\; =\; e^z\; \backslash left(\backslash Gamma\; (0,z)\; +\; \backslash gamma\; +\; \backslash log\; z\backslash right).$

Additional series representations for the r-order harmonic number exponential generating functions for integers $r\; \backslash geq\; 2$ are formed as special cases of these negative-order derivative-based series transformation results. For example, the second-order harmonic numbers have a corresponding exponential generating function expanded by the series

:$\backslash sum\_\{n\; \backslash geq\; 0\}\; \backslash frac\{H\_n^\{(2)\}\}\{n!\}\; z^n\; =\backslash sum\_\{j\; \backslash geq\; 1\}\; \backslash frac\{H\_j^2+H\_j^\{(2)\}\}\{2\; \backslash cdot\; (j+1)!\}\; z^j\; e^z\; \backslash left(j+1+z\backslash right).$

A further generalization of the negative-order series transformations defined above is related to more Hurwitz-zeta-like, or Lerch-transcendent-like, generating functions. Specifically, if we define the even more general parametrized Stirling numbers of the second kind by

:$\backslash left\backslash \{\backslash begin\{matrix\}\; k+2\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\_\{(\backslash alpha,\; \backslash beta)^\{\backslash ast\}\}\; :=\; \backslash frac\{1\}\{j!\}\; \backslash times\; \backslash sum\_\{0\; \backslash leq\; m\; \backslash leq\; j\}\; \backslash binom\{j\}\{m\}\; \backslash frac\{(-1)^\{j-m\}\}\{(\backslash alpha\; m+\backslash beta)^\{k\}\}$,

for non-zero $\backslash alpha,\; \backslash beta\; \backslash in\; \backslash mathbb\{C\}$ such that $-\backslash frac\{\backslash beta\}\{\backslash alpha\}\; \backslash notin\; \backslash mathbb\{Z\}^\{+\}$, and some fixed $k\; \backslash geq\; 1$, we have that

:$\backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{f\_n\}\{(\backslash alpha\; n+\backslash beta)^\{k\}\}\; z^n\; =\; \backslash sum\_\{j\; \backslash geq\; 1\}\; \backslash left\backslash \{\backslash begin\{matrix\}\; k+2\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\_\{(\backslash alpha,\; \backslash beta)^\{\backslash ast\}\}\; z^j\; F^\{(j)\}(z).$

Moreover, for any integers $u,\; u\_0\; \backslash geq\; 0$, we have the partial series approximations to the full infinite series in the previous equation given by

:$\backslash sum\_\{n=1\}^\{u\}\; \backslash frac\{f\_n\}\{(\backslash alpha\; n+\backslash beta)^\{k\}\}\; z^n\; =\; [w^u]\backslash left(\backslash sum\_\{j=1\}^\{u+u\_0\}\; \backslash left\backslash \{\backslash begin\{matrix\}\; k+2\; \backslash \backslash \; j\; \backslash end\{matrix\}\; \backslash right\backslash \}\_\{(\backslash alpha,\; \backslash beta)^\{\backslash ast\}\}\; \backslash frac\{(wz)^j\; F^\{(j)\}(wz)\}\{1-w\}\backslash right).$

Series for special constants and zeta-related functions resulting from these generalized derivative-based series transformations typically involve the generalized r-order harmonic numbers defined by

$H\_n^\{(r)\}(\backslash alpha,\; \backslash beta)\; :=\; \backslash sum\_\{1\; \backslash leq\; k\; \backslash leq\; n\}\; (\backslash alpha\; k\; +\; \backslash beta)^\{-r\}$ for integers $r\; \backslash geq\; 1$. A pair of particular series expansions for the following constants when $n\; \backslash in\; \backslash mathbb\{Z\}^\{+\}$ is fixed follow from special cases of BBP-type identities as

:$$

\begin{align}

\frac{4 \sqrt{3} \pi}{9} & = \sum_{j \geq 0} \frac{8}{9^{j+1}}\left(2 \binom{j+\frac{1}{3}}{\frac{1}{3}}^{-1} + \frac{1}{2} \binom{j+\frac{2}{3}}{\frac{2}{3}}^{-1}\right) \\

\log\left(\frac{n^2-n+1}{n^2}\right) & = \sum_{j \geq 0} \frac{1}{(n^2+1)^{j+1}}\left(\frac{2}{3 \cdot (j+1)} - n^2 \binom{j+\frac{1}{3}}{\frac{1}{3}}^{-1} + \frac{n}{2} \binom{j+\frac{2}{3}}{\frac{2}{3}}^{-1}\right).

\end{align}

Several other series for the zeta-function-related cases of the Legendre chi function, the polygamma function, and the Riemann zeta function include

:$$

\begin{align}

\chi_1(z) & = \sum_{j \geq 0} \binom{j+\frac{1}{2}}{\frac{1}{2}}^{-1} \frac{z \cdot (-z^2)^j}{(1-z^2)^{j+1}} \\

\chi_2(z) & = \sum_{j \geq 0} \binom{j+\frac{1}{2}}{\frac{1}{2}}^{-1} \left(1 + H_j^{(1)}(2, 1)\right) \frac{z \cdot (-z^2)^j}{(1-z^2)^{j+1}} \\

\sum_{k \geq 0} \frac{(-1)^k}{(z+k)^2} & = \sum_{j \geq 0} \binom{j+z}{z}^{-1} \left(\frac{1}{z^2} + \frac{1}{z} H_j^{(1)}(2, z)\right) \frac{1}{2^{j+1}} \\

\frac{13}{18} \zeta(3) & = \sum_{i=1,2} \sum_{j \geq 0} \binom{j+\frac{i}{3}}{\frac{i}{3}}^{-1} \left(\frac{1}{i^3} + \frac{1}{i^2} H_j^{(1)}(3, i) + \frac{1}{2i}\left(H_j^{(1)}(3, i)^2+H_j^{(2)}(3, i)\right)\right) \frac{(-1)^{i+1}}{2^{j+1}}.

\end{align}

Additionally, we can give another new explicit series representation of the inverse tangent function through its relation to the Fibonacci numbersSee equation (30) on the [http://mathworld.wolfram.com/InverseTangent.html MathWorld page] for the inverse tangent function. expanded as in the references by

:$\backslash tan^\{-1\}(x)\; =\; \backslash frac\{\backslash sqrt\{5\}\}\{2\backslash imath\}\; \backslash times\; \backslash sum\_\{b\; =\; \backslash pm\; 1\}\; \backslash sum\_\{j\; \backslash geq\; 0\}\; \backslash frac\{b\}\{\backslash sqrt\{5\}\}\; \backslash binom\{j+\backslash frac\{1\}\{2\}\}\{j\}^\{-1\}\backslash left[$

\frac{\left(b\imath\varphi t / \sqrt{5}\right)^j}{\left(1-\frac{b\imath\varphi t}{\sqrt{5}}\right)^{j+1}} -

\frac{\left(b\imath\Phi t / \sqrt{5}\right)^j}{\left(1+\frac{b\imath\Phi t}{\sqrt{5}}\right)^{j+1}}\right],

for $t\; \backslash equiv\; 2x\; /\; \backslash left(1+\backslash sqrt\{1+\backslash frac\{4\}\{5\}\; x^2\}\backslash right)$ and where the golden ratio (and its reciprocal) are respectively defined by $\backslash varphi,\backslash Phi\; :=\; \backslash frac\{1\}\{2\}\backslash left(1\; \backslash pm\; \backslash sqrt\{5\}\backslash right)$.

An inversion relation is a pair of equations of the form

:$g\_n\; =\; \backslash sum\_\{k=0\}^\{n\}\; A\_\{n,k\}\; \backslash cdot\; f\_k\; \backslash quad\backslash longleftrightarrow\backslash quad$

f_n = \sum_{k=0}^{n} B_{n,k} \cdot g_k,

which is equivalent to the orthogonality relation

:$\backslash sum\_\{k=j\}^\{n\}\; A\_\{n,k\}\; \backslash cdot\; B\_\{k,j\}\; =\; \backslash delta\_\{n,j\}.$

Given two sequences, $\backslash \{f\_n\backslash \}$ and $\backslash \{g\_n\backslash \}$, related by an inverse relation of the previous form, we sometimes seek to relate the OGFs and EGFs of the pair of sequences by functional equations implied by the inversion relation. This goal in some respects mirrors the more number theoretic (Lambert series) generating function relation guaranteed by the Möbius inversion formula, which provides that whenever

:$a\_n\; =\; \backslash sum\_\{d\; |\; n\}\; b\_d\; \backslash quad\backslash longleftrightarrow\backslash quad$

b_n = \sum_{d | n} \mu\left(\frac{n}{d}\right) a_d,

the generating functions for the sequences, $\backslash \{a\_n\backslash \}$ and $\backslash \{b\_n\backslash \}$, are related by the Möbius transform given by

:$\backslash sum\_\{n\; \backslash geq\; 1\}\; a\_n\; z^n\; =\; \backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{b\_n\; z^n\}\{1-z^n\}.$

Similarly, the Euler transform of generating functions for two sequences, $\backslash \{a\_n\backslash \}$ and $\backslash \{b\_n\backslash \}$, satisfying the relation{{cite web|last1=Weisstein|first1=E.|title=Euler Transform|url=http://mathworld.wolfram.com/EulerTransform.html|website=MathWorld}}

:$1\; +\; \backslash sum\_\{n\; \backslash geq\; 1\}\; b\_n\; z^n\; =\; \backslash prod\_\{i\; \backslash geq\; 1\}\; \backslash frac\{1\}\{(1-z^i)^\{a\_i\}\},$

is given in the form of

:$1\; +\; B(z)\; =\; \backslash exp\backslash left(\backslash sum\_\{k\; \backslash geq\; 1\}\; \backslash frac\{A(z^k)\}\{k\}\backslash right),$

where the corresponding inversion formulas between the two sequences is given in the reference.

The remainder of the results and examples given in this section sketch some of the more well-known generating function transformations provided by sequences related by inversion formulas (the binomial transform and the Stirling transform), and provides several tables of known inversion relations of various types cited in Riordan's Combinatorial Identities book. In many cases, we omit the corresponding functional equations implied by the inversion relationships between two sequences (this part of the article needs more work).

{{expand section|Need to add functional equations between generating functions related by the inversion pairs in the next subsections. For example, by exercise 5.71 of Concrete Mathematics, if $s\_n\; =\; \backslash sum\_\{k\; \backslash geq\; 0\}\; \backslash binom\{n+k\}\{m+2k\}\; a\_k$, then $S(z)\; =\; \backslash frac\{z^m\}\{(1-z)^\{m+1\}\}\; A\backslash left(\backslash frac\{z\}\{(1-z)^2\}\backslash right)$|date=March 2017}}

The first inversion relation provided below implicit to the binomial transform is perhaps the simplest of all inversion relations we will consider in this section. For any two sequences, $\backslash \{f\_n\backslash \}$ and $\backslash \{g\_n\backslash \}$, related by the inversion formulas

:$g\_n\; =\; \backslash sum\_\{k=0\}^\{n\}\; \backslash binom\{n\}\{k\}\; (-1)^k\; f\_k\; \backslash quad\backslash longleftrightarrow\backslash quad$

f_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k g_k,

we have functional equations between the OGFs and EGFs of these sequences provided by the binomial transform in the forms of

:$G(z)\; =\; \backslash frac\{1\}\{1-z\}\; F\backslash left(\backslash frac\{-z\}\{1-z\}\backslash right)$

and

:$\backslash widehat\{G\}(z)\; =\; e^z\; \backslash widehat\{F\}(-z).$

For any pair of sequences, $\backslash \{f\_n\backslash \}$ and $\backslash \{g\_n\backslash \}$, related by the Stirling number inversion formula

:$g\_n\; =\; \backslash sum\_\{k=1\}^\{n\}\; \backslash left\backslash \{\backslash begin\{matrix\}\; n\; \backslash \backslash \; k\; \backslash end\{matrix\}\; \backslash right\backslash \}\; f\_k\; \backslash quad\backslash longleftrightarrow\backslash quad$

f_n = \sum_{k=1}^{n} \left[\begin{matrix} n \\ k \end{matrix} \right] (-1)^{n-k} g_k,

these inversion relations between the two sequences translate into functional equations between the sequence EGFs given by the Stirling transform as

:$\backslash widehat\{G\}(z)\; =\; \backslash widehat\{F\}\backslash left(e^z-1\backslash right)$

and

:$\backslash widehat\{F\}(z)\; =\; \backslash widehat\{G\}\backslash left(\backslash log(1+z)\backslash right).$

These tables appear in chapters 2 and 3 in Riordan's book providing an introduction to inverse relations with many examples, though which does not stress functional equations between the generating functions of sequences related by these inversion relations. The interested reader is encouraged to pick up a copy of the original book for more details.

The terms, $A\_\{n,k\}$ and $B\_\{n,k\}$, in the inversion formulas of the form

:$a\_n\; =\; \backslash sum\_k\; A\_\{n,k\}\; \backslash cdot\; b\_k\; \backslash quad\backslash longleftrightarrow\backslash quad\; b\_n\; =\; \backslash sum\_k\; B\_\{n,k\}\; \backslash cdot\; (-1)^\{n-k\}\; a\_k,$

forming several special cases of Gould classes of inverse relations are given in the next table.

For classes 1 and 2, the range on the sum satisfies $k\; \backslash in\; [0,\; n]$, and for classes 3 and 4 the bounds on the summation are given by $k=n,n+1,\backslash ldots$. These terms are also somewhat simplified from their original forms in the table by the identities

:$\backslash binom\{p+qn-k\}\{n-k\}-q\; \backslash times\; \backslash binom\{p+qn-k-1\}\{n-k-1\}\; =\; \backslash frac\{p+qk-k\}\{p+qn-k\}\; \backslash binom\{p+qn-k\}\{n-k\}$

:$\backslash binom\{p+qk-k\}\{n-k\}+q\; \backslash times\; \backslash binom\{p+qk-k\}\{n-1-k\}\; =\; \backslash frac\{p+qn-n+1\}\{p+qk-n+1\}\; \backslash binom\{p+qk-k\}\{n-k\}.$

The so-termed simpler cases of the Chebyshev classes of inverse relations in the subsection below are given in the next table.

The formulas in the table are simplified somewhat by the following identities:

:$$

\begin{align}

\binom{n-k}{k} + \binom{n-k-1}{k-1} & = \frac{n}{n-k} \binom{n-k}{k} \\

\binom{n}{k} - \binom{n}{k-1} & = \frac{n+1-k}{n+1-2k} \binom{n}{k} \\

\binom{n+2k}{k}-\binom{n+2k}{k-1} & = \frac{n+1}{n+1+k} \binom{n+2k}{k} \\

\binom{n+k-1}{k} - \binom{n+k-1}{k-1} & = \frac{n-k}{n+k} \binom{n+k}{k}.

\end{align}

Additionally the inversion relations given in the table also hold when $n\; \backslash longmapsto\; n+p$ in any given relation.

The terms, $A\_\{n,k\}$ and $B\_\{n,k\}$, in the inversion formulas of the form

:$a\_n\; =\; \backslash sum\_k\; A\_\{n,k\}\; \backslash cdot\; b\_\{n+ck\}\; \backslash quad\backslash longleftrightarrow\backslash quad\; b\_n\; =\; \backslash sum\_k\; B\_\{n,k\}\; \backslash cdot\; (-1)^\{k\}\; a\_\{n+ck\},$

for non-zero integers $c$

forming several special cases of Chebyshev classes of inverse relations are given in the next table.

Additionally, these inversion relations also hold when $n\; \backslash longmapsto\; n+p$ for some $p=0,1,2,\backslash ldots,$ or when the sign factor of $(-1)^k$ is shifted from the terms $B\_\{n,k\}$ to the terms $A\_\{n,k\}$. The formulas given in the previous table are simplified somewhat by the identities

:$$

\begin{align}

\binom{n+ck+k}{k}-(c+1)\binom{n+ck+k-1}{k-1} & = \frac{n}{n+ck+k} \binom{n+ck+k}{k} \\

\binom{n}{k} + (c+1) \binom{n}{k-1} & = \frac{n+1+ck}{n+1-k} \binom{n}{k} \\

\binom{n-1+k}{k} + c \binom{n-1+k}{k-1} & = \frac{n+ck}{n} \binom{n-1+k}{k} \\

\binom{n+ck}{k} - (c-1) \binom{n+ck}{k-1} & = \frac{n+1}{n+1+ck-k} \binom{n+ck}{k}.

\end{align}

The Legendre–Chebyshev classes of inverse relations correspond to inversion relations of the form

:$a\_n\; =\; \backslash sum\_k\; A\_\{n,k\}\; \backslash cdot\; b\_k\; \backslash quad\backslash longleftrightarrow\backslash quad\; b\_n\; =\; \backslash sum\_k\; B\_\{n,k\}\; \backslash cdot\; (-1)^\{n-k\}\; a\_k,$

where the terms, $A\_\{n,k\}$ and $B\_\{n,k\}$, implicitly depend on some fixed non-zero $c\; \backslash in\; \backslash mathbb\{Z\}$. In general, given a class of Chebyshev inverse pairs of the form

:$a\_n\; =\; \backslash sum\_k\; A\_\{n,k\}\; \backslash cdot\; b\_\{n-ck\}\; \backslash quad\backslash longleftrightarrow\backslash quad\; b\_n\; =\; \backslash sum\_k\; B\_\{n,k\}\; \backslash cdot\; (-1)^\{k\}\; a\_\{n-ck\},$

if $c$ a prime, the substitution of $n\; \backslash longmapsto\; cn+p$, $a\_\{cn+p\}\; \backslash longmapsto\; A\_n$, and $b\_\{cn+p\}\; \backslash longmapsto\; B\_n$ (possibly replacing $k\; \backslash longmapsto\; n-k$) leads to a Legendre–Chebyshev pair of the formSee section 2.5 of Riordan

:$A\_n\; =\; \backslash sum\_k\; A\_\{cn+p,k\}\; B\_\{n-k\}\; \backslash quad\backslash longleftrightarrow\backslash quad\; B\_n\; =\; \backslash sum\_k\; B\_\{cn+p,k\}\; (-1)^k\; A\_\{n-k\}.$

Similarly, if the positive integer $c\; :=\; d\; e$ is composite, we can derive inversion pairs of the form

:$A\_n\; =\; \backslash sum\_k\; A\_\{dn+p,k\}\; B\_\{n-ek\}\; \backslash quad\backslash longleftrightarrow\backslash quad\; B\_n\; =\; \backslash sum\_k\; B\_\{dn+p,k\}\; (-1)^k\; A\_\{n-ek\}.$

The next table summarizes several generalized classes of Legendre–Chebyshev inverse relations for some non-zero integer $c$.

Abel inverse relations correspond to Abel inverse pairs of the form

:$a\_n\; =\; \backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; A\_\{nk\}\; b\_k\; \backslash quad\backslash longleftrightarrow\backslash quad\; b\_n\; =\; \backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; B\_\{nk\}(-1)^\{n-k\}\; a\_k,$

where the terms, $A\_\{nk\}$ and $B\_\{nk\}$, may implicitly vary with some indeterminate summation parameter $x$.

These relations also still hold if the binomial coefficient substitution of $\backslash binom\{n\}\{k\}\; \backslash longmapsto\; \backslash binom\{n+p\}\{k+p\}$ is performed for some non-negative integer $p$.

The next table summarizes several notable forms of these Abel inverse relations.

If we let the convolved Fibonacci numbers, $f\_k^\{(\backslash pm\; p)\}$, be defined by

:$$

\begin{align}

f_n^{(p)} & = \sum_{j \geq 0} \binom{p+n-j-1}{n-j} \binom{n-j}{j} \\

f_n^{(-p)} & = \sum_{j \geq 0} \binom{p}{n+j} \binom{n-j}{j} (-1)^{n-j},

\end{align}

we have the next table of inverse relations which are obtained from properties of ordinary sequence generating functions proved as in section 3.3 of Riordan's book.

Note that relations 3, 4, 5, and 6 in the table may be transformed according to the substitutions $a\_\{n-k\}\; \backslash longmapsto\; a\_\{n-qk\}$ and $b\_\{n-k\}\; \backslash longmapsto\; b\_\{n-qk\}$ for some fixed non-zero integer $q\; \backslash geq\; 1$.

Let $B\_n$ and $E\_n$ denote the Bernoulli numbers and Euler numbers, respectively, and suppose that the sequences, $\backslash \{d\_\{2n\}\backslash \}$, $\backslash \{e\_\{2n\}\backslash \}$, and $\backslash \{f\_\{2n\}\backslash \}$ are defined by the following exponential generating functions:See section 3.4 in Riordan.

:$$

\begin{align}

\sum_{n \geq 0} \frac{d_{2n} z^{2n}}{(2n)!} & = \frac{2z}{e^z-e^{-z}} \\

\sum_{n \geq 0} \frac{e_{2n} z^{2n}}{(2n)!} & = \frac{z^2}{e^z+e^{-z}-2} \\

\sum_{n \geq 0} \frac{f_{2n} z^{2n}}{(2n)!} & = \frac{z^3}{3(e^z-e^{-z}-2z)}.

\end{align}

The next table summarizes several notable cases of inversion relations obtained from exponential generating functions in section 3.4 of Riordan's book.Compare to the inversion formulas given in section 24.5(iii) of the NIST Handbook.

The inverse relations used in formulating the binomial transform cited in the previous subsection are generalized to corresponding two-index inverse relations for sequences of two indices, and to multinomial inversion formulas for sequences of $j\; \backslash geq\; 3$ indices involving the binomial coefficients in Riordan.See section 3.5 in Riordan's book.

In particular, we have the form of a two-index inverse relation given by

:$a\_\{mn\}\; =\; \backslash sum\_\{j=0\}^m\; \backslash sum\_\{k=0\}^n\; \backslash binom\{m\}\{j\}\; \backslash binom\{n\}\{k\}\; (-1)^\{j+k\}\; b\_\{jk\}\; \backslash quad\backslash longleftrightarrow\backslash quad$

b_{mn} = \sum_{j=0}^m \sum_{k=0}^n \binom{m}{j} \binom{n}{k} (-1)^{j+k} a_{jk},

and the more general form of a multinomial pair of inversion formulas given by

:$a\_\{n\_1\; n\_2\backslash cdots\; n\_j\}\; =\; \backslash sum\_\{k\_1,\; \backslash ldots,\; k\_j\}\; \backslash binom\{n\_1\}\{k\_1\}\; \backslash cdots\; \backslash binom\{n\_j\}\{k\_j\}\; (-1)^\{k\_1+\backslash cdots+k\_j\}$

b_{k_1 k_2 \cdots k_j} \quad\longleftrightarrow\quad

b_{n_1 n_2\cdots n_j} = \sum_{k_1, \ldots, k_j} \binom{n_1}{k_1} \cdots \binom{n_j}{k_j} (-1)^{k_1+\cdots+k_j}

a_{k_1 k_2 \cdots k_j}.

{{Reflist|30em}}

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• [https://youtube.com/watch?v=4AuV93LOPcE Why don't they teach Newton's calculus of 'What comes next?' - Mathologer]

Category:Generating functions