(Confusingly, the Pochhammer symbol that many use for "falling" factorials is used in special functions for "rising" factorials.) Similarly, the rising factorial, defined as formula_10, is a polynomial in "x" of degree "n" whose expansion is  with unsigned Stirling numbers of the first kind as coefficients. One expansion can be derived from the other by observing that formula_12. Stirling numbers of the second kind express reverse relations:  and  Considering the set of polynomials in the (indeterminate) variable "x" as a vector space, each of the three sequences is a basis. That is, every polynomial in "x" can be written as a sum formula_16 for some unique coefficients formula_17 (similarly for the other two bases). The above relations then express the change of basis between them, as summarized in the following commutative diagram: The coefficients for the two bottom changes are described by the Lah numbers below. Since coefficients in any basis are unique, one can define Stirling numbers this way, as the coefficients expressing polynomials of one basis in terms of another, that is, the unique numbers relating formula_18 with falling and rising factorials as above. Falling factorials define, up to scaling, the same polynomials as binomial coefficients: formula_19. The changes between the standard basis formula_20 and the basis formula_21 are thus described by similar formulas:  The Stirling numbers of the first and second kinds can be considered inverses of one another:  and  where formula_26 is the Kronecker delta. These two relationships may be understood to be matrix inverse relationships. That is, let "s" be the lower triangular matrix of Stirling numbers of the first kind, whose matrix elements formula_27 The inverse of this matrix is "S", the lower triangular matrix of Stirling numbers of the second kind, whose entries are formula_28 Symbolically, this is written  Although "s" and "S" are infinite, so calculating a product entry involves an infinite sum, the matrix multiplications work because these matrices are lower triangular, so only a finite number of terms in the sum are nonzero. Expressing a polynomial in the basis of falling factorials is useful for calculating sums of the polynomial evaluated at consecutive integers. Indeed, the sum of a falling factorial is simply expressed as another falling factorial (for "k≠-1")  This is analogous to the integral formula_31, though the sum should be over integers "i" strictly less than "n".