Struve function

The Struve functions are defined by $$\mathbf{H}_{\nu}(z)=\left(\dfrac{z}{2}\right)^{\nu+1} \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k\left(\frac{z}{2}\right)^{2k}}{\Gamma(k+\frac{3}{2})\Gamma(k+\nu+\frac{3}{2})}$$

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  Struve functions from Abramowitz&Stegun.

Properties

Theorem: If $x >0$ and $\nu \geq \dfrac{1}{2}$, then $\mathbf{H}_{\nu}(x) \geq 0$.

Proof: █

Relationship between Struve function and hypergeometric pFq

Theorem: The Struve function $H_n$ solves the following nonohomogeneous Bessel differential equation $$x^2y(x)+xy'(x)+(x^2-n^2)y(x)=\dfrac{4(\frac{x}{2})^{n+1}}{\sqrt{\pi}\Gamma(n+\frac{1}{2})}.$$

Proof: █

Relationship between Weber function 0 and Struve function 0
Relationship between Weber function 1 and Struve function 1

References

Struve functions in Abramowitz&Stegun