On the Pearson type VII distribution and the singular integral transformation

Abstract

It is shown that if m>0\[ \frac{x}{(1+x^2)m+1/2}=\lime→ +0∫|u|≥e\frac{1}{[1+(x-u)2]m+1/2}km(u)du\]holds for all x in the pointwise covergence and if m>½ this equality also holds in the Lp norm convergence (p≥1), where km(x) is a singular integral kernel, that is\[ km(x)=(sgn \ x)2π-2∫∞0e^{-|x|v}\frac{dv}{v(J2m(v)+Y2m(v))}.\]This equality is an extension of the well-known equality \[ \frac{x}{1+x2}=\lime → +0∫|u|≥e\frac{1}{1+(x-u)2}\frac{1}{π u}du.\]