3.2.1 Ord Family

The family of distributions has its origin in 1895 when Pearson introduced a system bearing his name in the continuous case. A discrete version of the Pearson system, defined by the difference equation

$f(x)-f(x-1)=\frac{(a-x)f(x-1)}{b_0+b_{1}x+b_{2}x(x-1)},$

was studied by Ord (1967a, 1967b), which is referred to as the Ord family in the sequel, where x takes values in a subset T of integers.

The members of the system with non-negative integers as the values of X are:

The first four moments of the family are given by

$\begin{array}{c}\hfill {\mu }_1^{\prime }=(b_1+a-1)/(1-2b_2),\hfill \\ \hfill {\mu }_2=(b_1+a-1)\{a{b}_2+(1-b_2)(b_1-2b_2)\}/{(1-2b_2)}^2(1-3b_2)\hfill \\ \hfill {\mu }_3={\mu }_2\{4b_2(a+1-b_1-b_2)-4b_1-3\}/(1-2b_2)(1-4b_2),\hfill \end{array}$

and

$\begin{array}{c}\hfill {\mu }_4=3{\mu }_2^2+\frac{{\mu }_2}{(1-4b_2)(1-5b_2)}[\{(1-b_2)(1-2b_2)\}\hfill \\ \hfill +6\{(b_1^2-b_2^2)+(b_2-b_1)-2b_2(a+b_1-1)\}\hfill \\ \hfill +b_2{\mu }_2\{1-2b_2{(1-3b_2)}^{-1}\}].\hfill \end{array}$

These will be required for analysing the descriptive characteristics of the distribution, estimating parameters by the method of moments, and in computing standard errors of statistics. All distributions of the system are either J shaped or unimodal. A crucial issue of interest when families are considered for model selection, is the criteria to distinguish its members. Ord (1972) considered the quantities $I=\frac{{\mu }_2}{\mu }$, which is the variance mean ratio called index of dispersion, and $S=\frac{{\mu }_3}{{\mu }_2}$, and proposed a diagram of $(S,I)$. In this diagram, the line $S=2I-1$ defines the binomial or Poisson or negative binomial distributions as $I<,=,>1$. The other distributions are obtained by analyzing $T=S-2I+1$. The region $S<2I-1$ gives the beta-binomial, $S>2I-1$ corresponds to the hypergeometric or beta-Pascal according to whether $S<1$ or >1 in addition. Since the parameters of the family are completely determined by the first three moments, the $(S,I)$ diagram can be used to locate a suitable distribution for a given data, by replacing the population moments with the corresponding sample moments. Ord (1972) also proposed the plot of $u_x=\frac{x{p}_x}{p_{x-1}}$ against x, where $p_x$ is the sample frequency. Since successive $u_x$ values are dependent, a smoothing $v_x=\frac{1}{2}(u_x+u_{x+1})$ is has also been suggested. For example, the binomial distribution satisfies

$u_x=\frac{(n+1)p}{q}-\frac{p}{q}x$

in the population, so that the plot of $(x,\frac{x{p}_x}{p_{x-1}})$ must show an approximate line. An interesting property of the Ord family is that the truncated version of X at either end also belongs to this family.

Various reliability functions, like the hazard rate, for the Ord family can not be expressed in simple forms. However, the members of the family admit a simple relationship between the hazard rate and the mean residual life. Ahmed (1991) has shown that X has binomial distribution in (3.2) if and only if

$m^{⁎}(x)=np+qxh(x).$

Since the mean residual life $m(x)=m^{⁎}(x)-x$, the relationship between $m(x)$ and $h(x)$ characterizing the binomial distribution is obvious. On similar lines, Osaki and Li (1988) have shown that X follows negative binomial distribution

$f(x)=\left(\begin{array}{c}n-1\\ x-1\end{array}\right)p^x{q}^{n-x},n⩾x,x=1,2,\dots ,$

if and only if

$E(X|X>m)=\mu +(m+1-x)h(m+1)/p$

for all integers $m⩾x-1$. They have also shown that X has Poisson distribution in (3.4) if and only if

$m^{⁎}(x)=\mu +\frac{(1+x)}{\lambda }h(x+1).$

A more general result for the Ord family in (3.1) rewritten as

$f(x+1)-f(x)=-\frac{(x+d)f(x)}{b_0+b_{1}x+b_2{x}^2},$

was characterized by Nair and Sankaran (1991) through the identity

$m^{⁎}(x)=\mu +(a_0+a_{1}x+a_2{x}^2)h(x+1),$

where

$b_i=\frac{a_i}{2a_2+1}\text{ and }d=\frac{a_1-a_2-\mu }{2a_2+1},i=0,1,2.$

The values of $a_0,a_1$ and $a_2$ for the distributions in (3.2)–(3.7) are listed in Table 3.1.

Table 3.1

Values of a₀,a₁ and a₂ in (3.9) for the Ord family

Distribution	$a_0$	$a_1$	$a_2$
binomial	q	q	0
Poisson	1	1	0
negative binomial	$\frac{1}{p}$	$\frac{1}{p}$	0
hypergeometric	$\frac{1}{2+2n+2M-N}$	$\frac{N+2-M}{2+2n+2M-1}$	$\frac{N-M-n}{2+2n+2M-1}$
negative hypergeometric	$\frac{N-k}{N+1-M}$	$\frac{N-k-1}{N+1-M}$	$\frac{1}{M-N-1}$
beta-Pascal	$\frac{-B}{A+2K+1}$	$\frac{-1+B}{A+2K+1}$	$\frac{-1}{A+2K+1}$

Figs 3.1A, 3.1B and 3.1C show the hazard rates of negative binomial, hypergeometric and negative hypergeometric distributions for various values of the parameters.

Various distributional properties of the individual models along with results on inference and applications have been discussed in Johnson et al. (1992). Further, characterization problems of the family in terms of truncated moments have been addressed by Glanzel et al. (1984) and Glanzel (1987, 1991). Of particular interest is the result

$E(X^2|X⩾x)=P(x)E(X|x⩾x)+Q(x)$

in Glanzel (1991), where $P(x)$ and $Q(x)$ are polynomials of degree at most one, with real coefficients. Obviously, (3.10) leads to a property of the variance residual life in relation to the mean residual life.

There are two related families in connection with (3.1) that deserves special mention. One is the Katz (1965) system defined by

$\frac{f(x+1)}{f(x)}=\frac{\alpha +\beta x}{1+x},x=0,1,2\dots ,\alpha >0,\beta <1.$

It is easy to see that (3.11) is a special case of (3.1) when $b_0=b_1$, $b_2=0$. This restricted model contains the binomial ($\beta <0$), Poisson ($\beta =0$) and negative binomial ($0<\beta <1$) distribution as particular members. These members can be discriminated by performing a test of hypothesis $H_0:{\mu }_2=\mu$ against $H_1:{\mu }_2>\mu$ (${\mu }_2<\mu$), with $H_0$ if accepted gives Poisson, otherwise $H_1:{\mu }_2>\mu$ (${\mu }_2<\mu$) gives negative binomial (binomial). The test is based on the statistic

$Z=\frac{S^2-\overline{x}}{\overline{x}}$

which is closely approximated by a normal distribution $N(\frac{\beta }{1-\beta },\frac{2}{n})$, where n is the sample size, and $S^2$ and $\overline{x}$ are the variance and mean of the sample. Following (3.9), for the Katz family, we see that

$m^{⁎}(x)=\mu {(1-\beta )}^{-1}(1+x)h(x+1)$

is satisfied for all x, and conversely. Sudheesh and Nair (2010) established the following characteristic properties for the Ord and Katz families, respectively:

${\sigma }^2(x)=E(c_0+c_{1}X+c_2{X}^2|X>x)+(\mu -m(x))(m(x)-x-1),$

where

$c_0=\mu +\frac{b_0-b_1+b_2}{1-2b_2},c_1=\frac{b_1-1}{1-2b_2},c_2=\frac{b_2}{1-2b_2},$

and

${\sigma }^2(x)=\frac{\alpha +\beta m(x)+(\mu -m(x))(m(x)-x-1)}{(1-\beta )}.$

For example, in the Poisson case, we have

${\sigma }^2(x)=\lambda +(\lambda -m(x))(m(x)-x-1),$

or alternatively in terms of the variance and the hazard rate as

${\sigma }^2(x)=\lambda +(x+1)(x+1-\lambda )h(x+1)-{(x+1)}^2{h}^2(x+1).$

Let τ be the class of real-valued functions $C(X)$ of X. Then, for the Ord family, we have

$\underset{C(x)\in \tau }{\inf }\frac{V(C(X))}{E^2[(d_0+d_{1}X+d_2{X}^2)\mathrm{\Delta }C(X)]}=1,$

where $d_i=c_i{\sigma }^{-1}$ and Δ is the forward difference operator, $i=0,1,2$. In particular, for the Katz family, we have

$\underset{C(X)\in \tau }{\inf }\frac{{\sigma }^2(1-\beta )V[C(X)]}{E^2((\alpha +\beta X)\mathrm{\Delta }C(X))}=1$

provides the lower bound to the variance of a random function $C(X)$. Under some regularity conditions, Nair and Sudheesh (2008) showed that these bounds reduce to the Cramer-Rao and Chapman-Robbins inequalities. The moments of the Katz family are

$\begin{array}{c}\hfill \mu =\frac{\alpha }{1-\beta },{\mu }_2=\frac{\alpha }{{(1-\beta )}^2},\hfill \\ \hfill {\mu }_3={\mu }_2(1+\beta )\text{ and }{\mu }_4=3{\mu }_2^2+{\mu }_2\frac{{(\beta +2)}^2}{{(1-\beta )}^2}.\hfill \end{array}$

It has a probability generating function

$P(t)={\left(\frac{1-\beta t}{1-\beta }\right)}^{-\frac{\alpha }{\beta }}.$

An extension of the Ord family has been provided by Sindhu (2002) wherein the difference equation in (3.1) has the modified form

$\frac{f(x+1)-f(x)}{f(x)}=\frac{d_0+d_{1}x+d_2{x}^2}{k_0+k_{1}x+k_2{x}^2}$

with $d_0,d_1,d_2,k_0,k_1$ and $k_2$ as real constants. Besides containing all members of the Ord family, the system in (3.14) contains several other distributions such as the confluent hypergeometric distribution of Bhattacharya (1966), Borel-Tanner model (Tanner, 1953), and the Haight distribution (Haight, 1961).

The approach for relationships between reliability functions in reversed time is the same and very similar results emerge. Corresponding to (3.9), the identity between $r(x)$ and $\lambda (x)$ for the Ord family is

$r(x+1)=\mu -(a_0+a_{1}x+a_2{x}^2)\lambda (x)$

with $a_0=\mu +\frac{b_0-b_1+b_2}{1-2a_2}$, $a_1=\frac{b_1-1}{1-2b_2}$ and $a_2=\frac{b_2}{1-2b_2}$.

Gupta et al. (1997) have given a formula for computing the hazard rate as

$\frac{1}{h(x)}=\sum _{i=0}^{\infty }\prod _{u=x}^{x+i}\frac{f(u+1)}{f(u)}$

when the ratio of two successive probabilities are known. By utilizing of this, for the Katz family, we have

$h(x)={[1+\frac{x!(k-1)!}{(k+x-1)!}\{\left(\begin{array}{c}k+x\\ 1+x\end{array}\right)\beta +\cdots +\left(\begin{array}{c}k+x+j-1\\ x+j\end{array}\right){\beta }^j+\dots \}]}^{-1},$

where $k=\frac{\alpha }{\beta }$. An alternative expression is

$h(x)={[{}_{2}F_1(1,\frac{\alpha }{\beta }+x;1+x;\beta )]}^{-1},$

where ${}_{2}F_1$ is the hypergeometric function defined as

${}_{2}F_1(a,b,c;x)=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)2!}{x}^2+\dots ,$

with $c\ne 0,-1,-2,\dots$.

For an elaborate study on the orthogonal polynomials in the cumulative Ord family and their application to variance bounds, one may refer to the recent work of Afendras et al. (2017).

3.2.2 Power Series Family

The power series family of distributions were initially studied by Kosambi (1949) and Noack (1950). Any distribution that can be represented as

$f(x)=\frac{a(x){\theta }^x}{A(\theta )},x=0,1,\dots ,\theta >0,$

where $a(x)>0$ and $A(\theta )=\sum a(x){\theta }^x$, is said to be of power series form. In (3.18), $A(\theta )$ is called the series function. Sometimes, (3.18) is also referred to as the discrete linear exponential family in view of the specification of (3.18) as

$f(x)=a(x)b(\alpha )e^{-\alpha x}.$

Initially, Noack (1950) considered $f(x)$ with support as the whole set of non-negative integers. Patil (1962) extended the support to any arbitrary non-empty subset T of non-negative integers such that

$A(\theta )=\sum _{x\in T}a(x){\theta }^x,$

with $a(x)>0$, $\theta ⩾0$, so that $A(\theta )$ is positive, finite and differentiable. Then, (3.18) for $x\in T$ is called a generalized power series distribution (GPSD). The system in (3.19) includes the binomial, negative binomial, Poisson and logarithmic series distributions. A truncated GPSD is also a GPSD.

A further extension of the GPSD has been introduced by Gupta (1974) who defined the modified power series distributions (MPSD), specified by the probability mass function of the form

$f(x)=\frac{a(x){(g(\theta ))}^x}{B(\theta )},x\in T,$

where $a(x)>0$, and $g(\theta )$ and $B(\theta )$ are positive finite and differentiable. When $g(\theta )$ is invertible, MPSD reduces to the GPSD. Apart from all distributions belonging to the GPSD family, the modified version contains more distributions, such as the generalized binomial distribution of Jain and Consul (1971). Being the more general form, the properties of (3.20) will be mentioned so that the results for GPSD can be deduced from them. The moments of (3.20) are

$\begin{array}{cc}\hfill \mu & =E(X)=\frac{B^{\prime }(\theta )g(\theta )}{B(\theta )g^{\prime }(\theta )}=\frac{g(\theta )}{g^{\prime }(\theta )}\frac{d}{d\theta }[\log B(\theta )],\hfill \\ \hfill {\mu }_2& =\frac{g(\theta )}{g^{\prime }(\theta )}\frac{d\mu }{d\theta },\hfill \end{array}$

and

${\mu }_{r+1}=\frac{g(\theta )}{g^{\prime }(\theta )}\frac{d{\mu }_r}{d\theta }+r{\mu }_2{\mu }_{r-1},r=1,2,\dots$

The factorial moments can be calculated from the recurrence relation

${\mu }_{(r+1)}=\frac{g(\theta )}{g^{\prime }(\theta )}\frac{d{\mu }_{(r)}}{d\theta }+{\mu }_{(r)}\mu -r{\mu }_{(r)}.$

Consul (1995) has developed characterizations of the exponential family of distributions

$f(x)=\exp [xp(\alpha )+q(\alpha )+s(x)]$

which contains numerous discrete probability mass functions and probability density functions of continuous random variables for various choices of $p(\alpha )$, $q(\alpha )$ and $S(x)$. In the discrete case, the translations $a(x)=\exp [s(x)]$, $B(\theta )=\exp [-q(\alpha )]$ and $g(\theta )=\exp [p(\alpha )]$ reduce (3.21) to the form (3.20). His main result can be stated in our notation as

$m^{⁎}(x)=\mu +\frac{g(\theta )}{g^{\prime }(\theta )}\frac{\partial }{\partial \theta }(\log S(x+1)).$

As examples, the Lagrangian Poisson distribution

$\begin{array}{c}f(x)=\frac{{(1+\alpha x)}^{x-1}}{x!}{\theta }^x\exp [-\theta (1+\alpha x)],x=0,1,\dots ,\hfill \\ \hfill \theta >0,0⩽\alpha ⩽{\theta }^{-1},\end{array}$

satisfies the property

$m^{⁎}(x)=\frac{\theta }{1-\alpha \theta }+{(-\alpha +\frac{1}{\theta })}^{-1}\frac{\partial }{\partial \theta }\log S(x+1);$

the Lagrangian negative binomial with probability function

$\begin{array}{c}f(x)=\frac{n}{n+mx}\left(\begin{array}{c}n+mx\\ x\end{array}\right){\theta }^x{(1-\theta )}^{n+mx-x},x=0,1,2,\dots ,\hfill \\ \hfill 0<\theta <1,n>0,m=0,1⩽m⩽{\theta }^{-1},\end{array}$

is characterized by

$m^{⁎}(x)=\frac{n\theta }{1-m\theta }+\frac{\theta (1-\theta )}{1-m\theta }\frac{\partial }{\partial \theta }\log S(x+1),0<\theta <1.$

Notice that (3.24) contains, as special cases, the binomial distribution ($m=0$), negative binomial ($m=1$), the Geeta distribution

$f(x)=\frac{1}{\alpha x-1}\left(\begin{array}{c}\alpha x-1\\ x\end{array}\right){\theta }^{x-1}{(1-\theta )}^{\alpha x-x},x=0,1,2\dots ,$

when $n=\alpha -1$ and x is replaced by $x-1$ so that the identity in (3.25) is true for $x=1,2,\dots ,n$ (binomial), for all x (negative binomial), and $x=2,3,\dots$ (Geeta). The identity in (3.22) can be used to obtain the lower bound for the variance as

$V(C(X))⩾E^2[g(X)\mathrm{\Delta }C(X)],$

when $C(x)$ is any real-valued function of x (Nair and Sudheesh, 2008) and

$g(x)=\frac{g(\theta )}{g^{\prime }(\theta )}\frac{1}{\sigma f(x)}\frac{\partial S(x+1)}{\partial \theta }.$

3.2.3 Lerch Family

An important sub-family of the MPSD is the Hurwitz-Lerch zeta distributions (HLZD). Various properties of this class, along with their applications to reliability, have been studied by Gupta et al. (2008). The HLZD arises from the general Hurwitz-Lerch zeta function defined by

$\varphi (z,s,a)=\sum _{k=0}^{\infty }\frac{z^k}{{(k+a)}^s},s\in C\text{ when }|z|<1,a=C(0,-1,-2,\dots )$

and $R(s)>1$ when $|z|=1$ and C is the set of complex numbers. Some known special cases of the function (3.26) are the Riemann zeta function

$\zeta (s)=\sum _{k=1}^{\infty }\frac{1}{k^s}=\sum _{k=0}^{\infty }\frac{1}{{(1+k)}^s}=\varphi (1,s,1),$

Hurwitz zeta function

$\zeta (s,a)=\sum _{k=0}^{\infty }\frac{1}{{(k+a)}^s}=\varphi (1,s,a),$

and the polylogarithmic function

$A(z,s)=\sum _{k=1}^{\infty }\frac{z^k}{k^s}=z\sum _{k=0}^{\infty }\frac{z^k}{{(1+k)}^s}=z\varphi (z,s,1).$

For detailed properties of these functions, we refer to Erdelyi et al. (1953). Zornig and Altmann (1995) introduced the three parameter HLZD as a generalization of the Zipf, Zipf-Mandelbrot and Good distributions mentioned below. Subsequently, the family has found some applications as a model in ecology, linguistics, information sciences, statistical physics and survival analysis.

Aksenov and Savageau (2005) defined the Lerch family of distributions through the probability mass function

$f(x)=\frac{c{z}^x}{{(a+x)}^s},x=0,1,2,\dots ,$

where $c^{-1}=\varphi (z,s,a)$, the Lerch transcendent (Lerch, 1887), also called the Hurwitz-Lerch zeta function defined in (3.26) for $z>0$, $a>0$. By virtue of the relationship

$\varphi (z,s,a)=z^m\varphi (z,s,m+a)+\sum _{k=0}^{m-1}\frac{z^k}{{(k+a)}^s},$

we can arrive at the distribution function

$F(x)=1-z^{x+1}\varphi (z,s,a+x+1)/\varphi (z,s,a).$

The probability generating function and the moment generating function are, respectively,

$P(t)=\frac{\varphi (tz,s,a)}{\varphi (z,s,a)},|t|⩽1,$

and

$M(t)=\frac{\varphi (z{e}^t,s,a)}{\varphi (z,s,a)}.$

There is unimodality for the family if $s<0$ and $a⩾1$ with mode at

$m_0=1+[\frac{1}{z^{1/s}-1}-a],$

where $[W]$ is the integer part of W. Furthermore,

$\mu =\frac{\varphi (z,s-1,a)}{\varphi (z,s,a)}-a$

and

${\mu }_2={(a+\mu )}^2+\frac{\varphi (z,s-2,a)-2(a+\mu )\varphi (z,s-1,a)}{\varphi (z,s,a)}.$

Regarding the reliability functions, we obtain directly from the definitions that

$h(x)=\frac{1}{{(a+x)}^s\varphi (z,s,a+x)}$

and

$\lambda (x)=\frac{c{z}^x}{\varphi (z,s,a)-z^{x+1}\varphi (z,s,a+x+1)}.$

We observe that the expressions in (3.32) leads to some new interesting recurrence relationships and formulae. Since

$\varphi (z,s,a+x)={[h(x){(a+x)}^s]}^{-1}$

and

$\varphi (z,s,a+x+1)=\frac{1}{z}[\varphi (z,s,a+x)-{(a+x)}^{-s}]={[h(x+1){(a+x+1)}^s]}^{-1}$

from (3.31), we have on eliminating ϕ between the last two equations,

$h(x+1)=z{\left(\frac{a+x}{a+x+1}\right)}^s\frac{h(x)}{1-h(x)},$

a recurrence relation for evaluating $h(x)$. One may use the initial value $h(0)=c{a}^{-s}$ in (3.34) to initiate the recurrence.

The probability mass function and the hazard rate function of the family are presented in Figs 3.2 and 3.3 respectively.

To simplify (3.33) further, we apply (Magnus et al., 1966)

$\varphi (z,s,a)-z^{x+1}\varphi (z,s,ax+1)=\sum _{k=0}^x\frac{z^k}{{(a+k)}^s}$

to write a closed-form expression

$\begin{array}{cc}\hfill \lambda (x)& =\frac{z^x}{(a+x^s{\sum }_{k=0}^x\frac{z^k}{{(a+k)}^s})}\hfill \\ \hfill & ={\left[\sum _{k=0}^x{\left(\frac{a+x}{a+x-k}\right)}^s\frac{1}{z^k}\right]}^{-1}.\hfill \end{array}$

The mean residual life is given by

$m(x)=\frac{1}{z^x\varphi (z,s,a+x+1)}\sum _{t=x+1}^{\infty }{z}^t\varphi (z,s,a+t).$

It now follows that

$m(x+1)z\varphi (z,s,a+x+2)-m(x)\varphi (z,s,a+x+1)=\frac{z}{{(a+x+1)}^s},$

or

$m(x+1)-m(x)=\frac{z+m(x+1)}{{(a+x+1)}^s\varphi (z,s,a+x+1)}.$

Lerch family can be written in the form (3.20) with

$B(\theta )=\varphi (z,s,a),a(x)={(a+x)}^{-s}\text{ and }g(\theta )=z$

as a modified power series family with parameter z in place of $g(\theta )=\theta$. Hence, the moments satisfy the recurrence formula

${\mu }_{r+1}=z\frac{d{\mu }_r}{dz}+r{\mu }_z{\mu }_{r-1},r=1,2,\dots ,$

for the central moments, and

${\mu }_{(r+1)}=z\frac{d{\mu }_{(r)}}{dz}+{\mu }_{(r)}\mu -r{\mu }_{(r)}$

connecting the factorial moments. With the use of the expressions for μ and ${\mu }_2$ given above, the central and factorial moments can be computed. In the context of reliability modelling, the following characterization in terms of reliability concepts are useful.

The estimation of parameters of the Lerch family has been addressed by Aksenov and Savageau (2005), who proposed the method of moments and the method of maximum likelihood. Denoting the sample moments by

$p_r=\frac{1}{n}\sum _{\lambda =1}^n{x}_i^r$

from a random sample $(x_1,\dots ,x_n)$ from the distribution, the equations to be solved in the method of moments are as follows:

$\begin{array}{c}\hfill \frac{\varphi (\overline{z},\overline{s}-1,\overline{a})}{\varphi (\overline{z},\overline{s},\overline{a})}-\overline{a}=p_1,\hfill \\ \hfill {\overline{a}}^2-2\overline{a}\frac{\varphi (\overline{z},\overline{s}-1,\overline{a})}{\varphi (\overline{z},\overline{s},\overline{a})}+\frac{\varphi (\overline{z},\overline{s}-2,\overline{a})}{\varphi (\overline{z},\overline{s},\overline{a})}=p_2,\hfill \\ \hfill -{\overline{a}}^3+3{\overline{a}}^2\frac{\varphi (\overline{z},\overline{s}-1,\overline{a})}{\varphi (\overline{z},\overline{s},\overline{a})}-3\overline{a}\frac{\varphi (\overline{z},\overline{s}-3,\overline{a})}{\varphi (\overline{z},\overline{s},\overline{a})}=p_3,\hfill \end{array}$