the following recurrence relation: Powers of an OGF Let be fixed, suppose that , and denote . Then we have a series expansion for given by and the coefficients satisfy a recurrence relation of the form Another formula for the coefficients, , is expanded by the Bell polynomials as where denotes the Pochhammer symbol. Logarithms of an OGF If we let and define , then we have a power series expansion for the composite generating function given by where the coefficients, , 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: Faà di Bruno's formula Let denote the EGF of the sequence, , and suppose that is the EGF of the sequence, . Faà di Bruno's formula implies that the sequence, , generated by the composition , can be expressed in terms of the exponential Bell polynomials as follows: Integral transformations OGF ⟷ EGF conversion formulas We have the following integral formulas for which can be applied termwise with respect to when is taken to be any formal power series variable: 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 , defined in. 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 . Example: A double factorial integral for the EGF of the Stirling numbers of the second kind The single factorial function, , is expressed as a product of two double factorial functions of the form where an integral for the double factorial function, or rational gamma function, is given by for natural numbers . This integral representation of then implies that for fixed non-zero and any integral powers , we have the formula Thus for any prescribed integer ,