In mathematics, the theta operator is a
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
defined by
:
This is sometimes also called the homogeneity operator, because its
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s are the
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...
s in ''z'':
:
In ''n'' variables the homogeneity operator is given by
:
As in one variable, the
eigenspace
In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
s of ''θ'' are the spaces of
homogeneous function
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar (mathematics), scalar, then the function's value is multiplied by some p ...
s. (
Euler's homogeneous function theorem
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; ...
)
See also
*
Difference operator
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
*
Delta operator In mathematics, a delta operator is a shift-equivariant linear operator Q\colon\mathbb \longrightarrow \mathbb /math> on the vector space of polynomials in a variable x over a field \mathbb that reduces degrees by one.
To say that Q is shift-equi ...
*
Elliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which im ...
*
Fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
D f(x) = \frac f(x)\,,
and of the integration ...
*
Invariant differential operator In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on \mathbb^n, functions on a manifold, vector valued fun ...
*
Differential calculus over commutative algebras In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this ...
References
Further reading
*{{cite book, last=Watson, first=G.N., title=A treatise on the theory of Bessel functions, year=1995, publisher=Univ. Press, location=Cambridge, isbn=0521483913, edition=Cambridge mathematical library ed.,
achdr. der2.
Differential operators