is called an algebraic differential equation, which, in this case, has the solutions and — the Bessel functions of the first and second kind respectively. Hence, we say that and are differentially algebraic (also algebraically transcendental). Most of the familiar special functions of mathematical physics are differentially algebraic. All algebraic combinations of differentially algebraic functions are differentially algebraic. Furthermore, all compositions of differentially algebraic functions are differentially algebraic. Hölder’s Theorem simply states that the gamma function, , is not differentially algebraic and is therefore transcendentally transcendental.
Let , and assume that a non-zero polynomial exists such that
As a non-zero polynomial in can never give rise to the zero function on any non-empty open domain of (by the Fundamental Theorem of Algebra), we may suppose, without loss of generality, that contains a monomial term having a non-zero power of one of the indeterminates .
Assume also that has the lowest possible overall degree with respect to the lexicographic ordering . For example,
because the highest power of in any monomial term of the first polynomial is smaller than that of the second polynomial.
Next, observe that
If we define a second polynomial by the transformation
then we obtain the following algebraic differential equation for :
Furthermore, if is the highest-degree monomial term in , then the highest-degree monomial term in is . Consequently, the polynomial
has a smaller overall degree than , and as it clearly gives rise to an algebraic differential equation for , it must be the zero polynomial by the minimality assumption on . Hence, defining by , we get
Now, let in to obtain
A change of variables then yields , and an application of mathematical induction (along with a change of variables at each induction step) to the earlier expression
This is possible only if is divisible by , which contradicts the minimality assumption on . Therefore, no such exists, and so is not differentially algebraic.Q.E.D.