## 6 Functional Limits Associated with the Wright Functions  
In 1969, [PERSON] [8; 9; 10; 11; 12; 13; 14] proposed the following delta sequence for a representation of the shifted Dirac delta function:  
\[\delta(x-1)=\lim_{v\rightarrow\infty}[v\,J_{v}(vx)] \tag{66}\]  
As was demonstrated over the 2000-2008 period by [PERSON] [12; 17; 20], this delta sequence is useful in the evaluation of asymptotic relations, limits of series, integrals, and integral representations of elementary and special functions. If the Lamborn expression is multiplied by a function \(f(tx)\) and integrated from zero to infinity with respect to variable \(x\), we have  
\[f(t)=\int_{0}^{\infty}f(tx)\delta(x-1)\,dx=\lim_{v\rightarrow\infty}\bigg{[}v \,\int_{0}^{\infty}f(tx)J_{v}(v\,x)\,dx\bigg{]} \tag{67}\]  
In such a way, function \(f(t)\) is represented by the asymptotic limit of the infinite integral of product of \(f(tx)\) and the Bessel function \(J_{v}(vx)\). If the right-hand integral in (67) can be evaluated in closed form, then the limit can be regarded as the generalization of [PERSON] rule. \[ \begin{array}{l}f(t)=\lim_{v\rightarrow\infty}[v\,\Phi(t,v)]\\ \Phi(t,v)=\int_{0}^{\infty}f(tx)J_{v}(v\,x)\,dx\end{array}  \tag{68}\]For the Wright function treated as the generalized Bessel function  
\[f(t)=W_{1,\beta+1}(-\frac{t^{2}}{4})=\bigg{(}\frac{2}{t}\bigg{)}^{\beta}J_{\beta} (t) \tag{69}\]  
it follows from (68) and (69) that  
\[ \begin{array}{l}f(t)=\lim_{\
u\rightarrow\infty}\Big{\{}\
u\,\int_{0}^{ \infty}J_{\
u}(\
u\,x)\,(\frac{2}{tx})^{X}J_{\beta}(tx)\,dx\,\Big{\}}=W_{1, \beta+1}\Big{(}-\frac{t^{2}}{4}\Big{)}\\ \Phi(t,\
u,\beta)=\int_{0}^{\infty}x^{-\beta}J_{\
u}(\
u\,x)\,J_{\beta}(tx)\,dx \end{array}  \tag{70}\]  
However, the infinite integral in (70) is known [13]:  
\[\Phi(t,\
u,\mu)=\int_{0}^{\infty}x^{-\beta}J_{\
u}(\
u\,x)\,J_{\beta}(tx)\,dx= \bigg{(}\frac{t}{2}\bigg{)}^{\beta}\frac{1}{\
u}\,{}_{2}F_{1}\bigg{(}\frac{ \
u+1}{2},\frac{1-\
u}{2};\beta+1;\frac{t^{2}}{\
u^{2}}\bigg{)} \tag{71}\]  
Therefore, the Wright function is represented by the following limit:  
\[ \begin{array}{l}W_{1,\beta+1}\Big{(}-\frac{t^{2}}{4}\Big{)}=\lim_{\
u \rightarrow\infty}\Big{\{}{}_{2}F_{1}\bigg{(}\frac{\
u+1}{2},\frac{1-\
u}{2}; \beta+1;\frac{t^{2}}{\
u^{2}}\bigg{)}\Big{\}}\\ Re(\
u+1)>0\quad;\quad Re\beta>-1\quad;\quad 0<t<\
u\end{array}  \tag{72}\]  
or in the equivalent form of  
\[W_{1,\beta+1}(-\,x)=\lim_{\
u\rightarrow\infty}\bigg{\{}{}_{2}F_{1}\bigg{(} \frac{\
u+1}{2},\frac{1-\
u}{2};\beta+1;\frac{4\,x}{\
u^{2}}\bigg{)}\bigg{\}} \tag{73}\]  
For \(\beta=0\), we have  
\[W_{1,1}\bigg{(}-\frac{t^{2}}{4}\bigg{)}=J_{0}(t)=\lim_{\
u\rightarrow\infty} \bigg{\{}{}_{2}F_{1}\bigg{(}\frac{\
u+1}{2},\frac{1-\
u}{2};1;\frac{t^{2}}{ \
u^{2}}\bigg{)}\bigg{\}} \tag{74}\]  
and for \(\beta=\pm 1/2\). \[ \begin{array}{l}W_{1,3/2}\Big{(}-\frac{t^{2}}{4}\Big{)}=\sqrt{\frac{t}{2}}J _{1/2}(t)=\lim_{\
u\rightarrow\infty}\Big{\{}{}_{2}F_{1}\bigg{(}\frac{\
u+1} {2},\frac{1-\
u}{2};\frac{3}{2};\frac{t^{2}}{\
u^{2}}\bigg{)}\Big{\}}=\frac{2 \sin t}{\sqrt{\pi}-t}\\ W_{1,1/2}\Big{(}-\frac{t^{2}}{4}\Big{)}=\sqrt{\frac{t}{2}}J_{-1/2}(t)=\lim_{\
u \rightarrow\infty}\Big{\{}{}_{2}F_{1}\bigg{(}\frac{\
u+1}{2},\frac{1-\
u}{2}; \frac{1}{2};\frac{t^{2}}{\
u^{2}}\bigg{)}\Big{\}}=-\frac{\cos t}{\sqrt{\pi}-} \end{array}  \tag{75}\]  
Hypergeometric Functions (75) are known in a different form [14]:  
\[ \begin{array}{l}{}_{2}F_{1}\big{(}a,1-a;\frac{3}{2};(\sin z)^{2}\big{)}= \frac{\sin[(2a-1)z]}{(2a-1)\sin z}\\ {}_{2}F_{1}\Big{(}a,1-a;\frac{1}{2};(\sin z)^{2}\big{)}=\frac{\cos[(2a-1)z]}{ \cos z}\\ a=\frac{\
u+1}{2}\quad;\quad\sin z=\frac{t}{\
u}\end{array}  \tag{76}\]  
If the delta sequence in (66) is used together with integral transforms having different kernels \(T\), we have [12; 20]  
\[ \begin{array}{l}\lim_{\
u\rightarrow\infty}[\
u\,\int_{0}^{\infty}f(\xi, \lambda)\,T\{J_{\
u}(\
u\,x),\xi\}\,d\xi]=\\ \lim_{\
u\rightarrow\infty}[\
u\,\int_{0}^{\infty}J_{\
u}(\
u\,x)\,T\{J(\xi,\lambda),x\}\,dx]=T(1,\lambda)\end{array}