A Further Property of Functions in Class ${\mathbf{B}}^{(m)}$

We say that a function $\alpha (x)$ belongs to the set ${\mathbf{A}}^{(\gamma )}$ if it has an asymptotic expansion of the form $\alpha (x)\sim {\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_i{x}^{\gamma -i}$ as $x\to \mathrm{\infty }$, which can be differentiated term by term infinitely many times. A function $f(x)$ is in the class ${\mathbf{B}}^{(m)}$ if it satisfies a linear homogeneous differential equation of the form $f(x)={\sum }_{k=1}^m{p}_k(x)f^{(k)}(x)$, with $p_k\in {\mathbf{A}}^{(i_k)}$, $i_k$ being integers satisfying $i_k\le k$. These functions have been shown to have many interesting properties, and their integrals ${\int }_0^{\mathrm{\infty }}f(x)dx$, whether convergent or divergent, can be evaluated very efficiently via the Levin–Sidi $D^{(m)}$-transformation. (In case of divergence, they are defined in some summability sense, such as Abel summability or Hadamard finite part or a mixture of these two.) In this note, we show that if $f(x)$ is in ${\mathbf{B}}^{(m)}$, then so is $(f\circ g)(x)=f(g(x))$, where $g(x)>0$ for all large $x$ and $g\in {\mathbf{A}}^{(s)}$, $s$ being a positive integer. This enlarges the scope of the $D^{(m)}$-transformation considerably to include functions of complicated arguments. We demonstrate the validity of our result with an application of the $D^{(3)}$ transformation to two integrals $I[f]$ and $I[f\circ g]$, for some $f\in {\mathbf{B}}^{(3)}$ and $g\in {\mathbf{A}}^{(2)}$.