Delta operator

In mathematics, a delta operator is a shift-equivariant linear operator Q : K [ x ] ⟶ K [ x ] {displaystyle Qcolon mathbb {K} longrightarrow mathbb {K} } on the vector space of polynomials in a variable x {displaystyle x} over a field K {displaystyle mathbb {K} } that reduces degrees by one. In mathematics, a delta operator is a shift-equivariant linear operator Q : K [ x ] ⟶ K [ x ] {displaystyle Qcolon mathbb {K} longrightarrow mathbb {K} } on the vector space of polynomials in a variable x {displaystyle x} over a field K {displaystyle mathbb {K} } that reduces degrees by one. To say that Q {displaystyle Q} is shift-equivariant means that if g ( x ) = f ( x + a ) {displaystyle g(x)=f(x+a)} , then In other words, if f {displaystyle f} is a 'shift' of g {displaystyle g} , then Q f {displaystyle Qf} is also a shift of Q g {displaystyle Qg} , and has the same 'shifting vector' a {displaystyle a} . To say that an operator reduces degree by one means that if f {displaystyle f} is a polynomial of degree n {displaystyle n} , then Q f {displaystyle Qf} is either a polynomial of degree n − 1 {displaystyle n-1} , or, in case n = 0 {displaystyle n=0} , Q f {displaystyle Qf} is 0. Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in x {displaystyle x} that maps x {displaystyle x} to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition when K {displaystyle mathbb {K} } has characteristic zero, since shift-equivariance is a fairly strong condition. Every delta operator Q {displaystyle Q} has a unique sequence of 'basic polynomials', a polynomial sequence defined by three conditions: Such a sequence of basic polynomials is always of binomial type, and it can be shown that no other sequences of binomial type exist. If the first two conditions above are dropped, then the third condition says this polynomial sequence is a Sheffer sequence—a more general concept.