---
author:
- |
    NICOLAE COTFAS\
    Faculty of Physics\
    University of Bucharest\
    PO Box 76-54, Postal Office 76, Bucharest\
    Romania\
    ncotfas@yahoo.comhttp://fpcm5.fizica.unibuc.ro/[${}_{{}_{\tilde{}}}$]{}ncotfas\
       A hypergeometric type equation satisfying certain conditions defines either a finite or\
    an infinite system of orthogonal polynomials.    We present in a unified and explicit way all these\
    systems of orthogonal polynomials, the associated special functions and some systems of coherent\
    states.   This general formalism allows us to extend some results known only in particular cases.\
     Orthogonal polynomials, Associated special functions, Coherent states, Raising and\
    lowering operators, Creation and annihilation operators, Hypergeometric-type equations.  
title: '**Associated special functions and coherent states**'
---

[ **1  Introduction**]{}\
Many problems in quantum mechanics and mathematical physics lead to equations of the type $$\label{hypeq}
\sigma (s)y''(s)+\tau (s)y'(s)+\lambda y(s)=0$$ where $\sigma (s)$ and $\tau (s)$ are polynomials of at most second and first degree, respectively, and $\lambda $ is a constant. These equations are usually called [*equations of hypergeometric type*]{} \[15\], and each can be reduced to the self-adjoint form $$[\sigma (s)\varrho (s)y'(s)]'+\lambda \varrho (s)y(s)=0$$ by choosing a function $\varrho $ such that $$[\sigma (s)\varrho (s)]'=\tau (s)\varrho (s).$$

The equation (\[hypeq\]) is usually considered on an interval $(a,b)$, chosen such that $$\begin{array}{r}
\sigma (s)>0\qquad {\rm for\ all}\quad s\in (a,b)\\
\varrho (s)>0\qquad {\rm for\ all}\quad s\in (a,b)\\
\lim_{s\rightarrow a}\sigma (s)\varrho (s)
=\lim_{s\rightarrow b}\sigma (s)\varrho (s)=0.
\end{array}$$ Since the form of the equation (\[hypeq\]) is invariant under a change of variable $s\mapsto cs+d$, it is sufficient to analyse the cases presented in table 1. Some restrictions must be imposed on $\alpha $, $\beta $ in order for the interval $(a,b)$ to exist. We prove that equation (\[hypeq\]) defines an infinite sequence of orthogonal polynomials $$\Phi _0,\ \ \Phi _1,\ \ \Phi _2,\ ...$$ in the case $\sigma (s)\in \{ 1,\ s,\ 1-s^2\}$, and a finite one $$\Phi _0,\ \ \Phi _1,\ \ ...,\ \ \Phi _L$$ in the case $\sigma (s)\in \{ s^2-1,\ s^2,\ s^2+1\}$.

The literature discussing special function theory and its application to mathematical and theoretical physics is vast, and there are a multitude of different conventions concerning the definition of functions. A unified approach is not possible without a unified definition for the associated special functions. In this paper we define them as $$\Phi _{l,m}(s)=\left(\sqrt{\sigma (s)}\right)^m\, \frac{{\rm d}^m}{{\rm d}s^m}\Phi _l(s)$$ where $\Phi _l$ are the orthogonal polynomials defined by equation (\[hypeq\]). The table 1 allows one to pass in each case from our parameters $\alpha $, $\beta $ to the parameters used in different approach.

[*Table 1:*]{} Particular cases [(in each case $\tau (s)=\alpha s\!+\!\beta $).]{}

  --------------------------------------------------------------------------------------------------------------------------------
      $\begin{array}{l}                          $\varrho (s)$                             $\alpha ,\beta $           $(a,b) $
       \mbox{}\\[-2mm]                                                                                             
     \sigma (s)\\[-2mm]                                                                                            
           \mbox{}                                                                                                 
        \end{array}$                                                                                               
  ------------------------- -------------------------------------------------------- ----------------------------- ---------------
      $\begin{array}{l}            ${\rm e}^{\frac{1}{2}\alpha s^2+\beta s}$                  $\alpha <0$           $\mathbb{R}$
       \mbox{}\\[-1mm]                                                                                             
          1\\[-3mm]                                                                                                
           \mbox{}                                                                                                 
        \end{array}$                                                                                               

      $\begin{array}{l}                $s^{\beta -1} {\rm e}^{\alpha s}$                   $\begin{array}{l}        $(0,\infty )$
       \mbox{}\\[-3mm]                                                                      \alpha \!<\!0\\        
          s\\[-3mm]                                                                          \beta \!>\!0          
           \mbox{}                                                                           \end{array}$          
        \end{array}$                                                                                               

      $\begin{array}{l}                        $\begin{array}{l}                           $\begin{array}{l}          $(-1,1)$
       \mbox{}\\[-3mm]                          \mbox{}\\[-2mm]                          \alpha \!<\! \beta \\     
    1\!\!-\!\!s^2\\[-3mm]    (1\!+\!s)^{-\frac{\alpha -\beta }{2}-1}\times \\[1mm]     \alpha \!+\!\beta \!<\!0    
           \mbox{}           \times (1\!-\!s)^{-\frac{\alpha +\beta }{2}-1}\\[-2mm]          \end{array}$          
        \end{array}$                                \mbox{}                                                        
                                                  \end{array}$                                                     

      $\begin{array}{l}                        $\begin{array}{l}                           $\begin{array}{l}        $(1,\infty )$
       \mbox{}\\[-1mm]                          \mbox{}\\[-2mm]                       0\!<\! \alpha \!+\!\beta \\  
   s^2\!\!\!-\!\!1\\[-3mm]    (s\!+\!1)^{\frac{\alpha -\beta }{2}-1}\times \\[1mm]             \alpha <0           
           \mbox{}           \times (s\!-\!1)^{\frac{\alpha +\beta }{2}-1}\\[-2mm]           \end{array}$          
        \end{array}$                                \mbox{}                                                        
                                                  \end{array}$                                                     

      $\begin{array}{l}            $s^{\alpha -2}{\rm e}^{-\frac{\beta }{s}}$                      $                $(0,\infty )$
       \mbox{}\\[-3mm]                                                                     \begin{array}{l}        
         s^2\\[-3mm]                                                                       \alpha \!<\!0\\         
           \mbox{}                                                                           \beta \!>\!0          
        \end{array}$                                                                         \end{array}$          

      $\begin{array}{l}                        $\begin{array}{l}                              $\alpha <0$           $\mathbb{R}$
       \mbox{}\\[-1mm]                          \mbox{}\\[-2mm]                                                    
   s^2\!\!\!+\!\!1\\[-3mm]        (1+s^2)^{\frac{1}{2}\alpha -1}\times \\[1mm]                                     
           \mbox{}           \mbox{}\quad \times {\rm e}^{\beta \arctan s}\\[-2mm]                                 
        \end{array}$                                \mbox{}                                                        
                                                  \end{array}$                                                     
  --------------------------------------------------------------------------------------------------------------------------------

\[table\]

In our previous papers \[6, 7\], we presented a systematic study of the Schrödinger equations exactly solvable in terms of associated special functions following Lorente \[14\], Jafarizadeh and Fakhri \[12\]. In the present paper, our aim is to extend this unified formalism by including a larger class of creation/annihilation operators and some temporally stable coherent states of Gazeau-Klauder type \[1, 8–10, 13\].\
[ **2  Orthogonal polynomials\
    of hypergeometric-type**]{}\
Let $\tau (s)=\alpha s+\beta $ be a fixed polynomial, and let $$\lambda _l\!=-\frac{\sigma ''(s)}{2}l(l-1)-\tau '(s)l
\!=-\frac{\sigma ''}{2}l(l-1)-\alpha \,l$$ for any $l\in \mathbb{N}$. It is well-known \[15\] that for $\lambda =\lambda _l$, the equation (\[hypeq\]) admits a polynomial solution $\Phi _l=\Phi _l^{(\alpha ,\beta )}$ of at most $l$ degree $$\label{eq3}
\sigma (s) \Phi _l ''+\tau (s) \Phi _l '+\lambda _l\Phi _l=0.$$ If the degree of the polynomial $\Phi _l$ is $l$ then it satisfies the Rodrigues formula \[15\] $$\Phi _l(s)=\frac{B_l}{\varrho (s)}\frac{{\rm d}^l}{{\rm d}s^l}[\sigma ^l(s)\varrho (s)]$$ where $B_l$ is a constant. Based on the relation $$\{ \ \delta \in \mathbb{R}\ |\ 
\lim_{s\rightarrow a}\sigma (s)\varrho (s)s^\delta  
=\lim_{s\rightarrow b}\sigma (s)\varrho (s)s^\delta =0 \ \}$$ $$=\left\{
\begin{array}{lll}
[0,\infty ) & {\rm if} & \sigma (s)\in \{ 1,\ s,\ 1-s^2\}\\[2mm] 
[0,-\alpha ) & {\rm if} & \sigma (s)\in \{ s^2-1,\ s^2,\ s^2+1\}
\end{array} \right.$$ one can prove \[7\] that the system of polynomials $\{\Phi _l\ |\ l<\Lambda \}$, where $$\Lambda \!=\!\left\{ \begin{array}{lcl}
\infty & {\rm for} & \sigma (s)\in \{ 1,\ s,\ 1-s^2\}\\[2mm]
\frac{1-\alpha }{2} & { \rm for } & 
\sigma (s)\in \{ s^2\!-\!1,\ s^2,\ s^2\!+\!1\} 
\end{array}\right.$$ is orthogonal with weight function $\varrho (s)$ in $(a,b)$. This means that equation (\[hypeq\]) defines an infinite sequence of orthogonal polynomials $$\Phi _0,\ \ \Phi _1,\ \ \Phi _2,\ ...$$ in the case $\sigma (s)\in \{ 1,\ s,\ 1-s^2\}$, and a finite one $$\Phi _0,\ \ \Phi _1,\ \ ...,\ \ \Phi _L$$ with  $L=\max \{ l\in \mathbb{N}\ |\ l<(1-\alpha )/2\}$ in the case $\sigma (s)\in \{ s^2-1,\ s^2,\ s^2+1\}$.

The polynomials $\Phi _l^{(\alpha ,\beta )}$ can be expressed in terms of the classical orthogonal polynomials as [ $$\label{classical}
\begin{array}{l}
  \Phi _l^{(\alpha ,\beta )}(s)=\\[3mm]
\mbox{}\qquad \left\{ \begin{array}{lll}
H_l\left(\sqrt{\frac{-\alpha }{2}}\, s-\frac{\beta }{\sqrt{-2\alpha }}\right)  
& {\rm if} & \sigma (s)=1\\[2mm]
L_l^{\beta -1}(-\alpha s)  & {\rm if} & \sigma (s)=s\\[2mm]
P_l^{(-\frac{\alpha +\beta }{2}-1,\ \frac{-\alpha +\beta }{2}-1)}(s)  & {\rm if} & \sigma (s)=1\!\!-\!s^2\\[2mm]
P_l^{(\frac{\alpha -\beta }{2}-1,\ \frac{\alpha +\beta }{2}-1)}(-s)  & {\rm if} & \sigma (s)=s^2\!\!-\!1\\[2mm]
\left(\frac{s}{\beta }\right)^lL_l^{1-\alpha -2l}\left(\frac{\beta }{s}\right) 
& {\rm if} & \sigma (s)=s^2\\[2mm]
{\rm i}^lP_l^{(\frac{\alpha +{\rm i}\beta }{2}-1,\ \frac{\alpha -{\rm i}\beta }{2}-1)}({\rm i}s) 
& {\rm if} & \sigma (s)=s^2\!\!+\!1
\end{array} \right.
\end{array}$$ ]{} where $H_n$, $L_n^p $ and $P_n^{(p,q)}$ are the Hermite, Laguerre and Jacobi polynomials, respectively.\
[ **3  Associated special functions,\
    raising and lowering operators**]{}\
Let $l\in \mathbb{N}$, $l<\Lambda $, and let $m\in \{ 0,1,...,l\}$. The functions $$\label{def}
\Phi _{l,m}(s)=\kappa ^m(s)\frac{{\rm d}^m}{{\rm d}s^m}\Phi _l(s)$$ where $$\kappa (s)=\sqrt{\sigma (s)}$$ are called the [*associated special functions*]{}. If we differentiate (\[eq3\]) $m$ times and then multiply the obtained relation by $\kappa ^m(s)$ then we get the equation $$\label{Hm}
H_m \Phi _{l,m}=\lambda _l\Phi _{l,m}$$ where $H_m$ is the differential operator $$H_m =-\sigma (s) \frac{d^2}{ds^2}-\tau (s) \frac{d}{ds}
+\frac{m(m-2)}{4}\frac{(\sigma '(s))^2}{\sigma (s)} \ \ \ \ \ \mbox{}$$ $$\label{defHm}
 + \frac{m\tau (s)}{2}\frac{\sigma '(s)}{\sigma (s)}
-\frac{1}{2}m(m-2)\sigma ''(s)-m\tau '(s) .$$

The relation $$\label{scalarprod}
 \langle f,g\rangle 
=\int_a^b\overline{f(s)}\, g(s)\varrho(s)ds$$ defines a scalar product on the space $$\mathcal{H}_m={\rm span}\{ \Phi _{l,m} \ |\ m\leq l <\Lambda \}$$ spanned by $\{ \Phi _{l,m} \ |\ m\leq l <\Lambda \}.$ For each $m<\Lambda $, the special functions $\Phi _{l,m}$ with $m\leq l<\Lambda $ are orthogonal with weight function $\varrho (s)$ in $(a,b)$, and the functions corresponding to consecutive values of $m$ are related through the raising/lowering operators \[6, 7, 12\] $$\begin{array}{l}
A_m=\kappa (s)\frac{d}{ds}-m\kappa '(s)\\[3mm]
A_m^+=-\kappa (s)\frac{d}{ds}-\frac{\tau (s)}{\kappa (s)}-(m-1)\kappa '(s)
\end{array}$$ namely, $$\label{AmAm+}
\begin{array}{l}
A_m\Phi _{l,m}=\left\{ \begin{array}{lll}
0 & {\rm for} & l=m\\
\Phi _{l,m+1} & {\rm for} & m<l<\Lambda 
\end{array} \right. \\[5mm]
A_m^+\Phi _{l,m+1}\!=\!(\lambda _l\!-\!\lambda _m)\Phi _{l,m}\ \  
{\rm for}\ \ 0\leq m<l< \Lambda .
\end{array}$$ In addition, we have the relations \[5, 11\] $$\label{philm}
\Phi _{l,m}=
\frac{A_m^+ }{\lambda _l-\lambda _m}
\frac{A_{m+1}^+ }{\lambda _l-\lambda _{m+1}}...
\frac{A_{l-1}^+ }{\lambda _l-\lambda _{l-1}}\Phi _{l,l}$$ for  $0\leq m<l< \Lambda $, and $$\label{fact}
H_m-\lambda _m=A_m^+A_m\qquad H_{m+1}-\lambda _m
=A_mA_m^+$$ $$\label{interw}
H_mA_m^+=A_m^+H_{m+1}\qquad A_mH_m
=H_{m+1}A_m$$ for $m+1<\Lambda .$

The functions $$\phi _{l,m}=\Phi _{l,m}/||\Phi _{l,m}||$$ where $$||f||=\sqrt{\langle f,f\rangle }$$ are the [*normalized associated special functions*]{}. Since \[6, 7\] $$\label{norm}
||\Phi _{l,m+1}||
=\sqrt{\lambda _l-\lambda _m}\, ||\Phi _{l,m}||\qquad$$ they satisfy the relations $$\label{relations} \begin{array}{l}
A_m\phi _{l,m}\!=\!\left\{ \begin{array}{lll}
0 & {\rm for} & l=m\\
\sqrt{\lambda _l\!-\!\lambda _m}\phi _{l,m+1} & {\rm for} & m<l<\Lambda 
\end{array} \right.\\[3mm]
A_m^+\phi _{l,m+1}\!=\!\sqrt{\lambda _l\!-\!\lambda _m}\phi _{l,m}
\quad {\rm for}\  0\leq m<l<\Lambda \\[3mm]
\phi _{l,m}=
\frac{A_m^+ }{\sqrt{\lambda _l-\lambda _m}}
\frac{A_{m+1}^+ }{\sqrt{\lambda _l-\lambda _{m+1}}}...
\frac{A_{l-1}^+ }{\sqrt{\lambda _l-\lambda _{l-1}}}\phi _{l,l}. 
\end{array}$$\
[ **4  Coherent states in the\
    case $\sigma (s)\in \{ 1,\, s,\, 1-s^2\}$**]{}\
Let $m$ be a fixed natural number and $\gamma $ a fixed real number. The sequence $$\phi _{m,m},\quad \phi _{m+1,m},\quad \phi _{m+2,m},\ \dots$$ is a complete orthonormal sequence in the Hilbert space $$\mathcal{H}\!=\!\left\{ \varphi :(a,b)\longrightarrow \mathbb{C}\ \left| 
\ \int_a^b|\varphi (s)|^2\varrho (s)\, ds<\infty \right. \right\}$$ with scalar product (\[scalarprod\]), for any $m\in \mathbb{N}$.\
[**4.1  Creation and annihilation\
operators**]{}\
The linear operators (see figure 1) $$\begin{array}{l}
a_m,\ a_m^+:\mathcal{H}_m\longrightarrow  \mathcal{H}_m\\[3mm]
\mbox{}\qquad  a_m=U_m^{-1}A_m\qquad a_m^+=A_m^+U_m 
\end{array}$$ defined by using the unitary operator $$\begin{array}{l}
U_m:\mathcal{H}_m\longrightarrow  \mathcal{H}_{m+1}\\[3mm]
\mbox{}\qquad  U_m\phi _{l,m}={\rm e}^{-{\rm i}\gamma (\lambda _{l+1}-\lambda _l)}\phi _{l+1,m+1}
\end{array}$$ are mutually adjoint, $$\begin{array}{l}\label{repsu11}
a_m\phi _{l,m}\!\!=\!\!\left\{ \begin{array}{ll}
0 & {\rm for} \ l\!=\!m\\
\sqrt{\lambda _{l}\!-\!\!\lambda _m} {\rm e}^{{\rm i}\gamma (\lambda _l-\lambda _{l-1})} \phi _{l-1,m} &
{\rm for} \ l\!>\!m
\end{array} \right.\\[5mm]
a_m^+\phi _{l,m}\!\!=\!\!\sqrt{\lambda _{l+1}\!-\!\!\lambda _m} 
{\rm e}^{-{\rm i}\gamma (\lambda _{l+1}-\lambda _l)} \phi _{l+1,m}\   
{\rm for}  \  l\!\geq \!m
\end{array}$$ and $$H_m-\lambda _m=a_m^+a_m \qquad \qquad \ \mbox{}$$ $$\begin{array}{l}
[a_m^+,a_m]\phi _{l,m}=(\lambda _l-\lambda _{l+1})\phi _{l,m}\\[2mm]
\mbox{}\qquad \qquad \quad \ =(\sigma ''l+\alpha )\phi _{l,m}. 
\end{array}$$ Since the operator $R_m=[a_m^+,a_m]$ satisfies the relations $$[R_m,a_m^+]=\sigma ''a_m^+\qquad [R_m,a_m]=-\sigma ''a_m$$ the Lie algebra $\mathbb{L}_m$ generated by $\{ a_m^+,a_m\}$ is finite dimensional.\

(90,55)(12,0) (18.7,7)[$a_0 $]{} (11.8,7)[$a_0^+$]{} (28,18.3)[$A_0^+$]{} (28,12)[$A_0$]{} (24.9,8)[$U_0$]{} (18.7,22)[$a_0 $]{} (11.8,22)[$a_0^+$]{} (28,33.3)[$A_0^+$]{} (28,27)[$A_0$]{} (24.9,23)[$U_0$]{} (18.7,37)[$a_0 $]{} (11.8,37)[$a_0^+$]{} (28,48.3)[$A_0^+$]{} (28,42)[$A_0$]{} (24.9,38)[$U_0$]{} (43.7,22)[$a_1 $]{} (36.8,22)[$a_1^+$]{} (53,33.3)[$A_1^+$]{} (53,27)[$A_1$]{} (49.9,23)[$U_1$]{} (43.7,37)[$a_1 $]{} (36.8,37)[$a_1^+$]{} (53,48.3)[$A_1^+$]{} (53,42)[$A_1$]{} (49.9,38)[$U_1$]{} (68.7,37)[$a_2 $]{} (61.8,37)[$a_2^+$]{} (78,48.3)[$A_2^+$]{} (78,42)[$A_2$]{} (74.9,38)[$U_2$]{} (16,52)[$.$]{} (16,51)[$.$]{} (16,50)[$.$]{} (41,52)[$.$]{} (41,51)[$.$]{} (41,50)[$.$]{} (66,52)[$.$]{} (66,51)[$.$]{} (66,50)[$.$]{} (91,52)[$.$]{} (91,51)[$.$]{} (91,50)[$.$]{} (14.5,0)[$\phi _{0,0}$]{} (14.5,15)[$\phi _{1,0}$]{} (14.5,30)[$\phi _{2,0}$]{} (14.5,45)[$\phi _{3,0}$]{} (39.5,15)[$\phi _{1,1}$]{} (39.5,30)[$\phi _{2,1}$]{} (39.5,45)[$\phi _{3,1}$]{} (64.5,30)[$\phi _{2,2}$]{} (64.5,45)[$\phi _{3,2}$]{} (89.5,45)[$\phi _{3,3}$]{} (16,4)[(0,1)[9]{}]{} (18,13)[(0,-1)[9]{}]{} (22,15)[(1,0)[17]{}]{} (39,17)[(-1,0)[17]{}]{} (22,3)[(3,2)[16]{}]{} (16,19)[(0,1)[9]{}]{} (18,28)[(0,-1)[9]{}]{} (22,30)[(1,0)[17]{}]{} (39,32)[(-1,0)[17]{}]{} (22,18)[(3,2)[16]{}]{} (16,34)[(0,1)[9]{}]{} (18,43)[(0,-1)[9]{}]{} (22,45)[(1,0)[17]{}]{} (39,47)[(-1,0)[17]{}]{} (22,33)[(3,2)[16]{}]{} (41,19)[(0,1)[9]{}]{} (43,28)[(0,-1)[9]{}]{} (47,30)[(1,0)[17]{}]{} (64,32)[(-1,0)[17]{}]{} (47,18)[(3,2)[16]{}]{} (41,34)[(0,1)[9]{}]{} (43,43)[(0,-1)[9]{}]{} (47,45)[(1,0)[17]{}]{} (64,47)[(-1,0)[17]{}]{} (47,33)[(3,2)[16]{}]{} (66,34)[(0,1)[9]{}]{} (68,43)[(0,-1)[9]{}]{} (72,45)[(1,0)[17]{}]{} (89,47)[(-1,0)[17]{}]{} (72,33)[(3,2)[16]{}]{}

[**Theorem 1.**]{} [*The Lie algebra $\mathbb{L}_m$ is isomorphic*]{} $$\mbox{}\ \ to \ \ \left\{ \begin{array}{lcl}
h(2) & if & \sigma (s)\!\in \!\{ 1,\, s\}\\[2mm]
su(1,1) & if & \sigma (s)\!=\!1\!-\!s^2.
\end{array} \right.$$\
[*Proof*]{}. In the case $\sigma (s)\in \{ 1,s\}$ the operator $R_m$ is a constant operator, namely, $R_m=\alpha $. Since $\alpha <0$, the operators $P_+=\sqrt{-1/\alpha }\, a_m^+$, $P_-=\sqrt{-1/\alpha }\, a_m$ and the unit operator $I$ form a basis of $\mathbb{L}_m$ such that $$[P_+,P_-]=-I\qquad [I,P_\pm ]=0$$ that is, $\mathbb{L}_m$ is isomorphic to the Heisenberg-Weyl algebra $h(2)$. If $\sigma (s)=1-s^2$ then $K_+=a_m^+$, $K_-=a_m $ and $K_0=R_m$ form a basis of $\mathbb{L}_m$ such that $$[K_+,K_-]=-2 K_0 \qquad [K_0,K_\pm ]=\pm K_\pm . \qquad \quad \rule{3pt}{3pt}$$

The operator $a_m$ can be regarded as an annihilation operator, and $a_m^+$ as a creation operator.\
[**4.2  Coherent states**]{}\
Let $m\in \mathbb{N}$ be a fixed natural number. The functions  $|0\rangle $, $|1\rangle $, $|2\rangle $, $\cdots $ , where $$|n\rangle =\phi _{m+n,m}$$ satisfy the relations $$\begin{array}{l}
a_m|n\rangle =\left\{ \begin{array}{ll}
0 & {\rm if} \ \ n=0\\
\sqrt{e_n}\,{\rm e}^{{\rm i}\gamma (e_n-e_{n-1})} |n-1\rangle & {\rm if} \ \ n>0
\end{array} \right. \nonumber \\[5mm]
a_m^+|n\rangle = \sqrt{e_{n+1}}\, {\rm e}^{-{\rm i}\gamma (e_{n+1}-e_n)} |n+1\rangle \\[2mm]
(H_m-\lambda _m)|n\rangle =e_n|n\rangle \nonumber 
\end{array}$$ where $$\begin{array}{l}
e_n=\lambda _{m+n}-\lambda _m\\
\mbox{}\quad =\left\{ 
\begin{array}{lcl}
-\alpha n & {\rm if} & \sigma(s)\in \{ 1,s\}\\[2mm]
n(n+2m-\alpha -1) & {\rm if } & \sigma (s)=1-s^2 .
\end{array} \right.
\end{array}$$

For each $z\in \mathbb{C}$, the function $$|z, \gamma \rangle=\sum_{n=0}^\infty \frac{z^n\, 
{\rm e}^{-{\rm i}\gamma e_n}}{\sqrt{\varepsilon _n}}|n\rangle$$ with $$\varepsilon _n=\left\{ \begin{array}{lll}
1 & {\rm if} & n=0\\
e_1e_2...e_n & {\rm if } & n>0\, .
\end{array} \right.$$ is an eigenfunction of $a_m$ $$a_m|z, \gamma \rangle =z|z, \gamma \rangle$$ and $$\langle z, \gamma |z, \gamma \rangle 
\!=\!\left\{ \begin{array}{ll}
{\rm e}^{-\frac{|z|^2}{\alpha }}& {\rm if} \  \sigma (s)\!\in \!\{ 1,\, s\}\\[3mm]
\mbox{}_0F_1(2m\!-\!\alpha ;|z|^2) & {\rm if} \  \sigma (s)\!=\!1\!-\!s^2.
\end{array} \right.$$ where $$\mbox{}_0F_1(c;\!z)\!\!=\!\!1\!+\frac{1}{c}\frac{z}{1!}+\frac{1}{c(\!c\!+\!\!1\!)}\frac{z^2}{2!}
+\frac{1}{c(\!c\!+\!\!1\!)(\!c\!+\!\!2\!)}\frac{z^3}{3!}+\!\cdots$$ is the confluent hypergeometric function.\
By using the notation $z=r{\rm e}^{{\rm i}\theta }$ and the modified Bessel function $$K_\nu (z)=\frac{\pi }{2}\frac{I_{-\nu }(z)-I_\nu (z)}{{\rm sin}\, (\nu \pi )}$$ where $$I_\nu (z)=\sum_{n=0}^\infty \frac{\left(\frac{1}{2}z\right)^{\nu +2n}}
{n!\, \Gamma (\nu +n+1)}$$ we prove (following \[2\]) that $\{ \ |z, \gamma \rangle \ |\ z\in \mathbb{C}\}$ is a system of coherent states.\
[**Theorem 2.**]{} [ *The system of functions $$\{ \ |z, \gamma \rangle \ |\ z\in \mathbb{C}\}$$ satisfies the resolution of identity $$\int_\mathbb{C} d\mu \, |z, \gamma \rangle \langle z, \gamma |=I$$ for $d\mu $ defined as $$\label{last}
d\mu =\!\left\{ \begin{array}{l}
\frac{-1}{\pi \alpha }{\rm e}^{\frac{1}{\alpha }|z|^2}d({\rm Re}\, z)\, d({\rm Im}\, z )\\[2mm]
\mbox{}\qquad \qquad  \quad for \ \sigma (s)\!\in \!\{ 1, s\}\\[3mm]
\frac{2r^{2m-\alpha }}{\pi \Gamma (2m-\alpha )}
K_{\frac{\alpha +1}{2}-m}(2r)\, dr\, d\theta \\[2mm]
\mbox{}\qquad \qquad  \quad for \  \sigma (s)\!=\!1\!-\!s^2.
\end{array} \right.$$* ]{}\
[*Proof.*]{} If $\sigma (s)\in \{ 1,s\}$ then $$\label{cs1}
|z, \gamma \rangle =\sum_{n=0}^\infty
\frac{z^n\, {\rm e}^{-{\rm i}\gamma e_n}}{\sqrt{n!\, (-\alpha )^n}}|n\rangle .$$ By denoting $t=-\frac{r^2}{\alpha }$ and using the integration by parts we get $$\begin{array}{l} 
\frac{-1}{\pi \alpha }\!\int_\mathbb{C} \!d({\rm Re}\, z)\, 
d({\rm Im}\, z)|z,\!\gamma \rangle \langle z,\!\gamma |\\[5mm]
\mbox{}\  =\frac{-1}{\pi \alpha }\!\!\sum_{n,n'} {\rm e}^{-{\rm i}\gamma (e_n-e_{n'})}\times \\[5mm]
\times \!\!\left(\int_0^\infty \!\!{\rm e}^\frac{r^2}{\alpha }\!\frac{r^{n+n'+1}}
{\sqrt{n!\, n'!\, (-\alpha )^{n+n'}}}dr \!\!
\int_0^{2\pi }\!\!{\rm e}^{{\rm i}(n-n')\theta }
d\theta \! \right)\!|n\rangle \langle n'|\\[5mm]
\mbox{}\  =\frac{-2}{\alpha }\sum_n
\left( \int_0^\infty \!\!\!{\rm e}^\frac{r^2}{\alpha }\frac{1}{n!}
\left(\frac{r^2}{-\alpha }\right)^n\!\!r\, dr\!\right) |n\rangle \langle n|\\[5mm]
\mbox{}\  =\sum_n\left(\int_0^\infty \!\!\!\! {\rm e}^{-t}\frac{t^n}{n!}\, 
dt \right)|n\rangle \langle n| 
\!=\!\sum_n |n\rangle \langle n|=I.
\end{array}$$ If $\sigma (s)=1-s^2$ then $$\label{cs2}\begin{array}{l}
|z, \gamma \rangle =\sqrt{\Gamma (2m-\alpha )}
\sum_{n=0}^\infty \frac{z^n\, {\rm e}^{-{\rm i}\gamma e_n}}{\sqrt{n!\, \Gamma (n+2m-\alpha )}}|n\rangle 
\end{array}$$ Denoting $d\mu =\mu (r)\, dr\, d\theta $ we get $$\begin{array}{l}
\int_\mathbb{C} d\mu \, |z, \gamma \rangle \langle z, \gamma |\\[5mm]
=\sum_{n=0}^\infty \frac{2\pi\Gamma (2m-\alpha )}
{n!\, \Gamma (n+2m-\alpha )}
\left(\int_0^\infty r^{2n}\mu (r)\, dr \right) |n\rangle \langle n| 
\end{array}$$ and hence, we must have the relation (Mellin transformation) $$\label{mellin}
2\pi\Gamma (2m-\alpha )\!\!
\int_0^\infty \!\!\!r^{2n}\mu (r)\, dr \!=\! \Gamma (n+1)\, \Gamma (n+2m-\alpha ).$$ The formula \[4\] $$\int_0^\infty \!\!\!2x^{\eta +\xi }K_{\eta -\xi }(2\sqrt{x})\, x^{n-1}dx\!=\!
\Gamma (2\eta +n)\, \Gamma (2\xi +n)$$ for $x=r^2$,  $\eta =\frac{1}{2}$,  $\xi =m-\frac{\alpha }{2}$ becomes [ $$\label{mellin1}
4\!\!\!\int_0^\infty \!\!\!\!\!\!\!r^{2n}K_{\frac{\alpha +1}{2}-m}(2r)\, r^{2m-\alpha } dr
\!=\!\Gamma (n+1)\Gamma (n+2m-\alpha ).$$]{} The relations (\[mellin\]) and (\[mellin1\]) lead to (\[last\]).$\rule{3pt}{3pt}$\
If we consider the ‘number’ operator \[1, 8\] $$N:\mathcal{H}\longrightarrow \mathcal{H}\qquad N|n\rangle =n|n\rangle$$ that is, $$N=\sum_{n=0}^\infty n\, |n\rangle \langle n|$$ then the operator $H=H_m-\lambda _m$ can be written as $$\begin{array}{l}
H=\sum_{n=0}^\infty e_n|n\rangle \langle n|\\[5mm] 
\mbox{}\ \  =\!\left\{ \begin{array}{lll}
-\alpha N & {\rm if} & \sigma (s)\in \{ 1,\, s\}\\[3mm]
N(N+2m-\alpha -1) & {\rm if} & \sigma (s)=1-s^2.
\end{array} \right. 
\end{array}$$

The operators $a_m$ and $a_m^{\perp }$, where \[8\] $$a_m^{\perp }\!=\!\frac{N}{H}a_m^+ 
\!=\!\left\{ \begin{array}{lll}
-\frac{1}{\alpha }a_m^+ & {\rm if} & \sigma (s)\!\in \!\{ 1,\, s\}\\[3mm]
\frac{1}{N+2m-\alpha -1}a_m^+ & {\rm if} & \sigma (s)\!=\!1\!-\!s^2.
\end{array} \right.$$ satisfy the relations $$[a_m,a_m^{\perp }]\!=\!I\qquad [N,a_m^{\perp }]\!=\!a_m^{\perp }\qquad  [N,a_m]\!=\!-a_m.$$ Therefore, we can consider the non-unitary displacement operator \[8\] $$\begin{array}{l}
D(z)={\rm exp}(z\, a_m^{\perp }-\overline{z}\, a_m)\\[5mm]
\mbox{}\qquad ={\rm exp}\left( -\frac{1}{2}|z|^2\right)\ 
{\rm exp}(z\, a_m^{\perp })\ {\rm exp}(-\overline{z}\, a_m) 
\end{array}$$ and $$|z, \gamma \rangle =D(z)\, |0\rangle \qquad {\rm for\ any\ } z\in \mathbb{C}.$$

Since the Hermitian operators $$X=\frac{1}{\sqrt{2}}(a_m^++a_m) \qquad P=\frac{\rm i}{\sqrt{2}}(a_m^+-a_m)$$ satisfy the commutation relation $$[X,P]={\rm i}[a_m,a_m^+]$$ and $|z, \gamma \rangle $ are eigenstates of $a_m$, the coherent states $|z, \gamma \rangle $ minimize the uncertainty relation \[8\] $$(\Delta X)^2(\Delta P)^2\geq \frac{1}{4}\langle {\rm i}[X,P]\rangle ^2.$$

The presence of the phase factor in definition of $|z, \gamma \rangle $ leads to the temporal stability of these coherent states $${\rm e}^{-{\rm i}tH}|z, \gamma \rangle =|z, \gamma +t\rangle .$$\
[**4.3  Analytical representations**]{}\
The space $$\mathcal{F}_m\!=\!\left\{ f:\mathbb{C}\longrightarrow \mathbb{C} \left|
\begin{array}{l}
f \ {\rm is \ an \ analytic \ function}\\ 
\int_\mathbb{C}|f(z)|^2\, d\mu <\infty 
\end{array}
\right. \right\}$$ is a Hilbert space with the inner product $$(f,g)=\int_\mathbb{C}\overline{f(z)}\, g(z)\, d\mu$$ where $d\mu $ is the measure defined by (\[last\]).\
Following Bargmann \[3\], we associate to each\
$\varphi :(a,b)\longrightarrow \mathbb{C}$ from $\mathcal{H}$ $$\varphi =\sum_{n=0}^\infty c_n|n\rangle \qquad {\rm with}\qquad  
\sum_{n=0}^\infty |c_n|^2<\infty$$ the entire function $f:\mathbb{C}\longrightarrow \mathbb{C}$ $$f(z)\!=\!\langle \overline{z}, \gamma |\varphi \rangle 
\!=\!\!\sum_{n=0}^\infty \frac{z^n{\rm e}^{{\rm i}\gamma e_n}}{\sqrt{\varepsilon _n}}\langle n|\varphi \rangle
\!=\!\!\sum_{n=0}^\infty \frac{z^n{\rm e}^{{\rm i}\gamma e_n}}{\sqrt{\varepsilon _n}}c_n .$$ Particularly, the function $$u_n:\mathbb{C}\longrightarrow \mathbb{C}\qquad u_n(z)=\frac{z^n{\rm e}^{{\rm i}\gamma e_n}}{\sqrt{\varepsilon _n}}$$ corresponds to $\varphi =|n\rangle $, for any $n\in \mathbb{N}$.

From the relation (\[mellin\]) we get $$(z^n,z^k)=\varepsilon _n\, \delta _{nk}=\left\{
\begin{array}{ll}
\varepsilon _n & {\rm if}\ n=k\\
0 & {\rm if}\ n\not=k
\end{array}\right.$$ whence $$(f,g)=\sum_{n=0}^\infty \varepsilon _n\, \overline{c_n}\, d_n$$ for any two elements $f,g\in \mathcal{F}_m$ $$f=\sum_{n=0}^\infty c_nz^n\quad  {\rm and}\quad  
g=\sum_{n=0}^\infty d_nz^n .$$ For an entire function $f=\sum_{n=0}^\infty c_nz^n$ we have $$f\in \mathcal{F}_m \quad \Longleftrightarrow \quad \sum_{n=0}^\infty \varepsilon _n\, |c_n|^2<\infty .$$ Since $$(u_n,f)=\sqrt{\varepsilon _n}\,{\rm e}^{-{\rm i}\gamma e_n} c_n$$ the orthonormal system $u_0,\, u_1,\, u_2,\, ...$ is complete $$||f||^2=\sum_{n=0}^\infty \varepsilon _n\, |c_n|^2=\sum_{n=0}^\infty |(u_n,f)|^2.$$

The isometry $$\mathcal{H}\longrightarrow \mathcal{F}_m:\ \ 
\sum_{n=0}^\infty c_n|n\rangle \mapsto \sum_{n=0}^\infty c_n\, u_n$$ allows us to identify the two Hilbert spaces, and to get $$\begin{array}{l}
a_m\, u_n =\left\{ \begin{array}{ll}
0 & {\rm if} \ \ n=0\\
\sqrt{e_n}\,{\rm e}^{{\rm i}\gamma (e_n-e_{n-1})} u_{n-1} & {\rm if} \ \ n>0
\end{array} \right. \nonumber \\[5mm]
a_m^+\, u_n = \sqrt{e_{n+1}}\, {\rm e}^{-{\rm i}\gamma (e_{n+1}-e_n)} u_{n+1} \\[2mm]
R_m\, u_n=(e_n-e_{n+1})u_n
\end{array}$$ whence $$\begin{array}{l}
a_m\, z^n=e_n\, z^{n-1}\\
a_m^+\, z^n=z^{n+1}\\
R_m\, z^n=(e_n-e_{n+1})z^n .
\end{array}$$ From the relation $$e_n=\left\{ 
\begin{array}{lcl}
-\alpha n & {\rm if} & \sigma(s)\in \{ 1,s\}\\[2mm]
n(n+2m-\alpha -1) & {\rm if } & \sigma (s)=1-s^2 .
\end{array} \right.$$ one gets $$a_m=-\alpha \frac{d}{dz}\qquad a_m^+=z\qquad R_m=\alpha$$ in the case $\sigma(s)\in \{ 1,s\}$, and $$\begin{array}{l}
a_m=z\frac{d^2}{dz^2}+(2m-\alpha )\frac{d}{dz}\\[2mm]
a_m^+=z\\[2mm]
R_m=-2z\frac{d}{dz}-2m+\alpha 
\end{array}$$ in the case $\sigma (s)=1-s^2$.\
[**5  Coherent states in the\
    case $\sigma (s)\in \{ s^2-1,\, s^2,\, s^2+1\}$**]{}\
Let $\tau (s)=\alpha s+\beta $ be a fixed polynomial, $$L=\max \{ l\in \mathbb{N}\ |\ l<(1-\alpha )/2\}$$ and let $m$ be a fixed natural number with $0\leq m\leq L.$ The functions $|0\rangle $, $|1\rangle $, ..., $|\mathcal{L}\rangle $, where $\mathcal{L}=L-m$ and $$|n\rangle =\phi _{m+n,m}$$ are orthogonal and span a $(\mathcal{L}+1)$-dimensional space $$\mathcal{E}_m={\rm span}\{ |0\rangle , |1\rangle , ..., |\mathcal{L}\rangle \}.$$\
[**5.1  Creation and annihilation\
operators**]{}\
By following \[16\] and the analogy with the case $\sigma (s)\in \{ 1,\, s,\, 1-s^2\}$ we define the creation and annihilation operators $\tilde{a}_m^+,\, \tilde{a}_m:\mathcal{E}_m\longrightarrow \mathcal{E}_m$ $$\begin{array}{l}
\tilde{a}_m|n\rangle =\left\{ \begin{array}{ll}
0 & {\rm if} \ \ n=0\\
\sqrt{\tilde{e}_n}\,{\rm e}^{{\rm i}\gamma (e_n-e_{n-1})} |n-1\rangle & {\rm if} \ \ n>0
\end{array} \right. \nonumber \\[5mm]
\tilde{a}_m^+|n\rangle \!=\! \left\{ \begin{array}{ll}
\sqrt{\tilde{e}_{n+1}}\, {\rm e}^{-{\rm i}\gamma (e_{n+1}-e_n)} |n+1\rangle & {\rm if}\ \  n\!<\!\mathcal{L}\\
0 & {\rm if} \ \ n\!=\!\mathcal{L}
\end{array}\right. 
\end{array}$$ where $\tilde{e}_n=n(\mathcal{L}-n+1)$ and $$e_n=\lambda _{m+n}-\lambda _m=-n(n+2m+\alpha -1).$$

These operators satisfy the relations $$\begin{array}{l}
[\tilde{a}_m^+,\tilde{a}_m]=2\, \tilde{R}_m\\[2mm]
[\tilde{R}_m,\tilde{a}_m^+]=\tilde{a}_m^+\\[2mm]
[\tilde{R}_m,\tilde{a}_m]=-\tilde{a}_m
\end{array}$$ where $\tilde{R}_m$ is the operator $\tilde{R}_m=N-\frac{ \mathcal{L} }{2}$ defined by using the ‘number operator’ $$N:\mathcal{E}_m\longrightarrow \mathcal{E}_m\qquad N|n\rangle =n|n\rangle  .$$ Therefore, the Lie algebra generated by $\tilde{a}_m^+$ and $\tilde{a}_m$ is isomorphic to $su(2)$. The functions $|n\rangle $ are eigenfunctions of the operator $H=H_m-\lambda _m$ $$H|n\rangle =e_n|n\rangle .$$\
[**5.2  Coherent states**]{}\
By following \[16\] and the analogy with the case $\sigma (s)\in \{ 1,\, s,\, 1-s^2\}$, we consider for each $z\in \mathbb{C}$ the function $$|z, \gamma \rangle=\sum_{n=0}^\mathcal{L} \frac{z^n\, 
{\rm e}^{-{\rm i}\gamma e_n}}{\sqrt{\tilde{\varepsilon }_n}}|n\rangle$$ where $$\tilde{\varepsilon }_n=\left\{ \begin{array}{lll}
1 & {\rm if} & n=0\\
\tilde{e}_1\tilde{e}_2...\tilde{e}_n & {\rm if } & n>0\, .
\end{array} \right.$$ The system $\{ \ |z, \gamma \rangle \ |\ z\in \mathbb{C}\}$ is an overcomplete system of functions in the finite dimensional space $\mathcal{E}_m$ with $$\langle z, \gamma |z, \gamma \rangle =\sum_{n=0}^\mathcal{L}\frac{|z|^{2n}}{\tilde{\varepsilon }_n}$$ and the property (temporal stability) $${\rm e}^{-{\rm i}tH}|z, \gamma \rangle =|z, \gamma +t\rangle .$$ It can be regarded as a system of coherent states in $\mathcal{E}_m$.\
[**7  Concluding remarks**]{}\
The associated hypergeometric-type functions can be studied together in a unified formalism, and are directly related to the bound-state eigenfunctions of some important Schrödinger equations (Pöschl-Teller, Morse, Scarf, etc.).

It is useful to obtain fundamental versions (at the level of associated special functions) for some methods and formulae from quantum mechanics because in this way one can extend results known in particular cases to other quantum systems. A large number of formulae occurring in various applications of quantum mechanics follow from a very small number of fundamental mathematical results.\
[\
This work was supported by the grant CERES 4-192.]{}\
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