, are defined by the formal power series In this article, we use the convention that the ordinary (exponential) generating function for a sequence
Additionally, we use the bracket notation for coefficient extraction from the Concrete Mathematics reference which is given by
Series multisection provides formulas for generating functions enumerating the sequence
, another useful formula providing somewhat reversed floored arithmetic progressions are generated by the identity[2] The exponential Bell polynomials,
denote the coefficients in the expansion of the reciprocal generating function, then we have the following recurrence relation: Let
, in the previous expansion satisfy the recurrence relation given by and a corresponding formula expanded by the Bell polynomials in the form of the power series coefficients of the following generating function: Let
One particular formula results in the case of the double factorial function example given immediately below in this section.
, 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 which is convergent provided suitable conditions on the parameter
derivatives of the ordinary geometric series can be shown, for example by induction, to satisfy an explicit closed-form formula given by for any
EGF conversion formula cited above, we can compute the following corresponding exponential forms of the generating functions
: 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.
, we have that (compare to the special case of the integral formula for the Nielsen generalized polylogarithm function defined in[10]) [11] Notice that if we set
, we have the following integral representations for the so-termed square series generating function associated with the sequence
:[12] 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 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,
to obtain the result in the previous equation where the arithmetic progression generating function at hand is denoted by for each fixed
[13] The reference also provides nested coefficient extraction formulas of the form which are particularly useful in the cases where the component sequence generating functions,
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.
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.
denotes the triangle of first-order Eulerian numbers:[18] We can also expand the negative-order zeta series transformations by a similar procedure to the above expansions given in terms of the
in the table above), we have the following special case series for the dilogarithm and corresponding constant value of the alternating zeta function: When
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 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 for non-zero
, satisfying the relation[20] is given in the form of where the corresponding inversion formulas between the two sequences is given in the reference.
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).
The interested reader is encouraged to pick up a copy of the original book for more details.
forming several special cases of Chebyshev classes of inverse relations are given in the next table.
, be defined by 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.
are defined by the following exponential generating functions:[24] The next table summarizes several notable cases of inversion relations obtained from exponential generating functions in section 3.4 of Riordan's book.