An extended form of Bürmann's theorem. Let be a function of analytic in a ring-shaped region , bounded by another curve and an inner curve . Let be a function analytic on and inside having only one zero (which is simple) within the contour. Further let be a given point within . Finally, let(1)for all points of , and(2)for all points of . Then(3)where(4)(5)(Whittaker and Watson 1990, pp. 131-132).
The transform inverting the sequence(1)into(2)where the sums are over all possible integers that divide and is the Möbius function.The logarithm of the cyclotomicpolynomial(3)is closely related to the Möbius inversion formula.
Expresses a function in terms of its Radon transform,(1)(2)
Let be defined as a function of in terms of a parameter by(1)Then Lagrange's inversion theorem, also called a Lagrange expansion, states that any function of can be expressed as a power series in which converges for sufficiently small and has the form(2)The theorem can also be stated as follows. Let and where , then(3)(4)Expansions of this form were first considered by Lagrange (1770; 1868, pp. 680-693).
Let and be sequences of complex numbers such that for , and let the lower triangular matrices and be defined asandwhere the product over an empty set is 1. Then and are matrix inverses (Bhatnagar 1995, pp. 16-17).This result simplifies to the Gould and Hsu matrix inversion formula when , to Carlitz's -analog for (Carlitz 1972), and specialized to Bressoud's matrix theorem (Bressoud 1983) for and (Bhatnagar 1995, p. 17).The formula can also be extended to a summation theorem which generalizes Gosper's bibasic sum (Gasper and Rahman 1990, p. 240; Bhatnagar 1995, p. 19).
Let be a sequence of complex numbers and let the lower triangular matrices and be defined asandwhere the product over an empty set is 1. Then and are matrix inverses (Bhatnagar 1995, pp. 15-16 and 50-51). The Krattenthaler matrix inversion formula is a generalization of this result.
Bürmann's theorem deals with the expansion of functions in powers of another function. Let be a function of which is analytic in a closed region , of which is an interior point, and let . Suppose also that . Then Taylor's theorem gives the expansion(1)and, if it is legitimate to revert this series, the expression(2)is obtained which expresses as an analytic function of the variable for sufficiently small values of . If is then analytic near , it follows that is an analytic function of when is sufficiently small, and so there will be an expansion in the form(3)(Whittaker and Watson 1990, p. 129).The actual coefficients in the expansion are given by the following theorem, generally known as Bürmann's theorem (Whittaker and Watson 1990, p. 129). Let be a function of defined by the equation(4)Then an analytic function can, in a certain domain of values of , be expanded in the form(5)where the remainder term is(6)and is a contour..