Bessel functions of different orders orthogonality

Question

It is known that $$ \int_0^\infty\!J_m(ar)\,J_m(br)\,r\,dr = \frac{\delta(a-b)}{a} $$

Is there a similar equation for

$$ \int_0^\infty\!J_{m-1}(ar)\,J_{m+1}(br)\,r\,dr\;\text? $$

Are the Bessel functions of orders $m\!-\!1$ and $m\!+\!1$ orthogonal?

Answer 1

They are not orthogonal in general. Using the recurrence relationship

$$J_{n+1}(x)+J_{n-1}(x)=\frac{2n}{x}J_n(x)$$

we see that

$$\begin{align} \int_0^\infty J_{m-1}(ar)J_{m+1}(br)\,r\,dr&=\int_0^\infty \left(\frac{2m}{ar}J_m(ar)-J_{m+1}(ar)\right)J_{m+1}(br)\,r\,dr\\\\ &=\frac{2m}{a}\int_0^\infty J_m(ar)J_{m+1}(br)\,dr-\int_0^\infty J_{m+1}(ar) J_{m+1}(br)\,r\,dr\\\\ &=\frac{2m}{a}\int_0^\infty J_m(ar)J_{m+1}(br)\,dr-\frac{\delta(a-b)}{a}\tag 1 \end{align}$$

The first integral on the right-hand side of $(1)$ is not equal to $0$ in general. For example, with $m=1$, $a=2$, and $b=5$ its value is $\frac{2}{25}\ne 0$.