Here we introduce a new distribution that unifies both the COMNB and GCOMP distributions.
Definition 1. An rv X is said to follow the extended COMPoisson distribution with parameters (v, p, α, β) [ECOMP (v, p, α, β)] iff its pmf is given by
$$ P\left(X=k\right)=\frac{{\left\{{\left(\nu \right)}_k\right\}}^{\beta }}{{}_1S_{\alpha 1}^{\beta}\left(\nu;\;1;p\right)\;}\frac{p^k}{{\left(k!\right)}^{\alpha }}=\frac{{\left\{\Gamma \left(\nu +k\right)\right\}}^{\beta }}{{\left(\Gamma \nu \right)}^{\beta }{}_1S_{\alpha 1}^{\beta}\left(\nu;\;1;p\right)}\frac{p^k}{{\left(k!\right)}^{\alpha }} $$
(6)
The distribution is defined in the parameter space
$$ {\Theta}_{ECOM}=\left\{\nu \ge 0,\;p>0,\alpha >\beta \right\}\cup \left\{\nu >0,\;0<p<1,\;\alpha =\beta \right\}. $$
It may be noted that unlike in the COMNB distribution where the parameter α ≥ 1 and in the GCOMP distribution where the parameter β ≤ 1, in the ECOMP distribution these two parameters can be either positive or negative with the restriction of α ≥ β.
Particular cases: The ECOMP (ν, p, α, β) distribution reduces to COMNB (ν, p, α) for β = 1, to GCOMP (ν, p, β) for α = 1, to COMP (p, α − β) for ν = 1, to COMP (p, α) for β = 0, to Poisson (p) for ν = 1, α = β + 1, also to Poisson (p) for β = 0, α = 1, to NB (ν, p) for α = β = 1 and to a new generalization of NB(NGNB) distribution when α = β = γ with pmf
$$ P\left(X=k\right)={\left(\begin{array}{l}\nu +k1\\ {}\kern0.96em k\end{array}\right)}^{\gamma }{p}^k/{}_1S_{\gamma 1}^{\gamma}\left(\nu;\;1;p\right) $$
(7)
For 0 < ν ≤ 1, the distribution in (7) is logconvex as will be seen in proposition 4 in the Section 2.7.
2.1 Shape of the pmf
It is observed from the plots of the pmf of the ECOMP(v, p, α, β) distribution for different values of the parameters in Fig. 1, that the distribution is very flexible and can be non increasing with mode at zero, unique non zero mode, two modes and also bimodal with one mode always at zero.
2.2 Approximations of the normalizing constant
2.2.1 Approximation using truncation of the series
The normalizing constant \( {}_1S_{\alpha 1}^{\beta}\left(\nu; 1;p\right) \) of the ECOMP(v, p, α, β) distribution is not expressed in a closed form and includes the summation of infinite series. Therefore, we need approximations of this constant to compute the pmf and moments of the distribution numerically.
A simple approximation is to truncate the series, that is
$$ {}_1S_{\alpha 1,m}^{\beta}\left(\nu;\;1;p\right)={\displaystyle \sum_{k=0}^m\frac{{\left\{{\left(\nu \right)}_k\right\}}^{\beta }}{{\left(k\;!\right)}^{\alpha }}\kern0.24em {p}^k,} $$
(8)
where m is an integer chosen such that ε
_{
m
} = (ν − m + 1)^{β}
p/m
^{α} < 1. The relative truncation error is then given by the expression R
_{
m
}(ν, p, α, β) \( =\left\{{}_1S_{\alpha 1}^{\beta}\left(\nu;\;1;p\right){}_1S_{\alpha 1,m}^{\beta}\left(\nu;\;1;p\right)\;\right\}/{}_1S_{\alpha 1,m}^{\beta}\left(\nu;\;1;p\right)\;. \) Then the relative error about the pmf is give by {P
_{
m
}(k) − P(k)}/P(k), where P(k) is given by the right hand side (r.h.s.) of equation (6) in Section 2 and P
_{
m
}(k) is given by the r.h.s. of (6) with \( {}_1S_{\alpha 1}^{\beta}\left(\nu;\;1;p\right) \) substituted by \( {}_1S_{\alpha 1,m}^{\beta}\left(\nu;\;1;p\right) \). The upper bound of the relative truncation error is then found to be
$$ {R}_m\left(\nu,\;p,\alpha, \beta \right)<\frac{{\left\{{\left(\nu \right)}_{m+1}\right\}}^{\beta }{p}^{m+1}}{{\left\{\left(m+1\right)!\right\}}^{\alpha }{}_1S_{\alpha 1,m}^{\beta}\left(\nu;\;1;p\right)}{\displaystyle \sum_{k=0}^{\infty }{\varepsilon}_m^k}=\frac{{\left\{{\left(\nu \right)}_{m+1}\right\}}^{\beta }{p}^{m+1}}{{\left\{\left(1{\varepsilon}_m\right)\left(m+1\right)!\right\}}^{\alpha }{}_1S_{\alpha 1,m}^{\beta}\left(\nu;\;1;p\right)} $$
For α − β ≥ 1, this truncated approximation is good because ε
_{
m
} = O(1/m) and thus, the truncation point m is not large. However, for 0 < α − β < 1 and p > 1, the truncation point become too large to compute the approximation. For example, when ν = 1.5, p = 3, α = 3.1, β = 3, m has to be over 50,000. This is not practicable. To avoid this difficulty it is useful to make a restriction for the parameter p such that p < 1 when α − β → 0. For example, with the restriction p < 10^{α − β}, we see the relative truncation error R
_{50}(1.5, 3, 3.1, 3) < 0.001.
2.2.2 Asymptotic approximation of the normalizing constant using the Laplace’s method
It is also useful to consider an asymptotic approximation formula of the normalizing constant \( {}_1S_{\alpha 1}^{\beta}\left(\nu, 1,p\right) \). The approximation formula by the Laplace’s method (Bleistein and Handelsman 1986, Ch 8.3, pages 331–340) is given by
$$ {}_1S_{\alpha}^{\beta}\left(v;1,p\right)\approx \frac{p^{\left\{1\alpha +\left(2\nu 1\right)\beta \Big\}/2\Big(\alpha \beta \right)} \exp \left\{\left(\alpha \beta \right){p}^{1/\left(\alpha \beta \right)}\right\}}{{\left(2\pi \right)}^{\left(\alpha \beta 1\right)/2}\sqrt{\alpha \beta }{\left\{\Gamma (v)\right\}}^{\beta }} $$
(9)
This formula reduces to the asymptotic formula by Minka et al. (2003) when ν = 1 or β = 0 and that by Imoto (2014) when α = 1. The proof and numerical investigation about the formula (9) are given in Appendix A.1.
2.3 Recurrence relation for probabilities
The ECOMP (ν, p, α, β) pmf has a simple recurrence relation given by
$$ \frac{P\left(X=k+1\right)}{P\left(X=k\right)}=\frac{p{\left(\nu +k\right)}^{\beta }}{{\left(k+1\right)}^{\alpha }}\Rightarrow {\left(k+1\right)}^{\alpha }P\left(X=k+1\right)=p{\left(\nu +k\right)}^{\beta }P\left(X=k\right) $$
(10)
with \( P\left(X=0\right)={\left[{}_1S_{\alpha 1}^{\beta}\left(\nu;\;1;p\right)\;\right]}^{1} \). This will be useful for the computation of the probabilities. Further using (10) we can see that the ECOMP(v, p, α, β) distribution has a longer (shorter) tail than the COMNB(v, p, α) for α < (>)1 and a longer (shorter) tail than the GCOMP(v, p, β) for β > (<)1.
2.4 Exponential family
The pmf in (6) can also be expressed as
$$ P\left(X=k\right)= \exp \left[\beta\;\log {\left(\nu \right)}_k\beta\;\log \Gamma \left(\nu \right)\alpha\;\log k!+k\; \log p \log {}_1S_{\alpha 1}^{\beta}\left(\nu;\;1;p\right)\;\right] $$
(11)
Which immediately implies that the ECOMP (ν, p, α, β) distribution belongs to the exponential family with parameters ( log p, α, β) when v, is a nuisance parameter or when its value is given.
2.5 Index of dispersion
The pmf of ECOMP (ν, p, α, β) distribution in (6) can be seen as a weighted Poisson (p) distribution with weight function w(x) = {Γ(ν + x)}^{β}/(Γ(1 + x))^{α − 1}. As such it will be over (under) dispersed if w(x) in logconvex (logconcave). That is if \( \frac{d^2}{d{x}^2} \log \left[w(x)\right]\ge \left(\le \right)\;0 \). [See theorem 4 of Kokonendji et al. 2008]
$$ \begin{array}{l}\Rightarrow \beta \frac{d^2}{d{x}^2} \log \Gamma \left(\nu +x\right)+\left(1\alpha \right)\frac{d^2}{d{x}^2} \log \Gamma \left(1+x\right)\ge \left(\le \right)\;0\\ {}\Rightarrow \beta {\displaystyle \sum_{k\ge 0}\frac{1}{{\left(\nu +x+k\right)}^2}}\left(\alpha 1\right){\displaystyle \sum_{k\ge 0}\frac{1}{{\left(x+1+k\right)}^2}}\ge \left(\le \right)\;0\end{array} $$
[On using result 6.4.10 page 260 from Abramowitz and Stegun, 1970].
Hence, ECOMP (ν, p, α, β) is over dispersed (i) if α < 1, β ≥ 0 for all v (ii) if {α ≥ 1, β > 0} or {α < 1, β < 0} when {0 < ν ≤ 1, β ≤ α ≤ β + 1} or {ν > 1, α ≤ 1} and under dispersed (i) if α ≥ 1, β < 0 for all v (ii) for {α ≥ 1, β > 0} or {α < 1, β < 0} if {0 < ν ≤ 1, α ≥ β + 1} or {ν > 1, α ≥ 1}.
As a particular cases of the above result, when β = 1, we can see that the COMNB (ν, p, α) distribution always over dispersed for {0 < ν ≤ 1, 1 ≤ α < 2} or {ν > 1, α = 1} and under dispersed compared to COMP distribution for {0 < ν ≤ 1, α ≥ 2}. Similarly when α = 1, the GCOMP(v, p, β) distribution is seen to be is over dispersed for 0 < β ≤ 1 and under dispersed for β < 0. When ν = 1, we derive that COMP (p, α − β) is over dispersed for α − β > 1 under dispersed for α − β < 1 and equidispersed when α − β = 1. Finally, the new generalized NB distribution with pmf (7) is over dispersed when γ = 1 (which is when it reduces to Negative binomial) and under dispersed if γ > 1.
It can also be checked that ECOMP (v, p, α, β) is over (under) dispersed for α ≥ β > (≤)0 w.r.t. COMNB (v, p, α) and w.r.t. GCOMPoisson (v, p, β) it is over (under) dispersed for β ≤ α < 1 (1 < β ≤ α).
2.6 Different formulations of ECOMP (v, p, α, β)
Two different formulations of the proposed distribution are presented in this section.
2.6.1 ECOMP (v, p, α, β) as a distribution from a queuing set up
Like the COMPoisson distribution, the ECOMP (v, p, α, β) distribution can also be derived as the probability of the system being in the k
^{th} state for a queuing system with state dependent service and arrival rate.
Consider a single server queuing system with state dependent (that is dependent on the system state, k
^{th} state means k number of units in the system) arrival rate λ
_{
k
} = (ν + k)^{β}
λ, and state dependent service rate μ
_{
k
} = k
^{α}
μ , where, 1/μ and 1/λ are respectively the normal mean service and mean arrival time for a unit when that unit is the only one in the system; α and v are the pressure coefficients, reflecting the degree to which the service and arrival rates of the system are affected by the system state.
Proposition 1. Under the above set up where the arrival rate and the service rate increases exponentially as queue lengthens (i.e. as k increases) the probability of the system being in the k
^{th} state is ECOMP (v, p, α, β).
Proof: See Appendix B.1
2.6.2 ECOMP (v, p, α, β) as exponential combination formulation
The general form of the exponential combination of two pmfs say f
_{1}(x; θ
_{1}) and f
_{2}(x; θ
_{2}) is given by (Atkinson 1970)
$$ {\left\{{f}_1\left(x;{\theta}_1\right)\right\}}^{\beta }{\left\{{f}_2\left(x;{\theta}_2\right)\right\}}^{1\beta }/{\displaystyle \sum {f}_1{\left(x;{\theta}_1\right)}^{\beta }{f}_2{\left(x;{\theta}_2\right)}^{1\beta }} $$
This combining of the pmf was suggested by Cox (1961, 1962) for combining the two hypotheses (β = 1, i.e. the distribution is f
_{1} and β = 0 that is the distribution is f
_{2}) in a general model of which they would both be special cases. The inferences about β made in the usual way and testing the hypothesis that the value of β is zero or one is equivalent to testing for departures from one model in the direction of the other.
Proposition 2. ECOMP (ν, p, α, β) distribution is an exponential combination NB (v, λ) and COMPoisson (μ, θ) distributions, with λ
^{β}
μ
^{1 − β} = p and α = θ(1 − β) + β.
Proof: See Appendix B.2.
From the above formulations it is clear that for ECOMP (v, p, α, β),
β close to zero will indicate departure from COMPoisson towards NB, while β close to one will indicate the reverse. Thus ECOMP (v, p, α, β) can also be regarded as a natural extension of COMPoisson, and negative binomial distributions.
2.7 Logconcavity and modality
Proposition 3. The ECOMP (v, p, α, β) has a logconcave pmf when {ν > 1, p > 0, α ≥ β}
Proof: See Appendix B.3.
From the above result the corresponding results of COMNB (v, p, α) and GCOMP (v, p, β) can be obtained as particular cases. That is COMNB (v, p, α) is logconcave when {ν > 1, p > 0, α ≥ 1} and GCOMP (v, p, β) is logconcave when {ν > 1, p > 0, β ≤ 1}.
Following two important results follows as a consequence of logconcavity:
If {ν ≥ 1, p > 0, α > β} the ECOMP (v, p, α, β) distribution is
Using the recurrence relation of the probabilities in (10) it is observed that the ECOMP (ν, p, α, β) has

(i)
a non increasing pmf with a unique mode at X = 0 if ν
^{β}
p < 1,
e.g. ν = 2, α = 3, β = 2, p should be less than 0.25 to have unique mode at X = 0.

(ii)
a unique mode at X = k if k
^{α}/(ν + k − 1)^{β} < p < (k + 1)^{α}/(ν + k)^{β}
e.g. ν = 2, α = 3, β = 2, p should be between 1.6875 and 2.560 to have unique mode at X = 3.

(iii)
two modes at X = k and X = k − 1 if (ν + k − 1)^{β}
p = k
^{α}. In particular the two modes are at X = 0 and X = 1 if ν
^{β}
p = 1.
e.g. ν = 2, α = 3, β = 2, p should be equal to 4.408 to have two modes at X = 5 and X = 6.
Graphical illustrations of the above three examples are presented in the first plots of Fig. 1. It is interesting to note that the distribution may be bimodal with one of the mode always at zero as shown the last two plots in Fig. 1.
Proposition 4. ECOMP (v, p, α, β) has a logconvex pmf for {0 < ν ≤ 1, α = β}
Proof. See Appendix B.4.
Following important results follows as a consequence of logconvexity:
If {ν ≤ 1, p > 0, α = β} the ECOMP (v, p, α, β) distribution with pmf in (7)

➢ is Infinitely divisible (see Warde and Katti 1971) distribution, hence Discrete Compound Poisson distribution. (see page 409 of GómezDéniz et al. 2011)

➢ has an decreasing failure rate function, hence increasing mean residual life function

➢ has an upper bound for variance as p ν
^{β} (using result of page 410 of GómezDéniz et al. 2011)
2.8 Moments
The r
^{th} factorial moment E(X
^{[r]}) = μ
^{[r]} of the ECOMP (v, p, α, β) is given by
$$ \begin{array}{l}{\mu}^{\left[r\right]}=\frac{{\left\{{\left(\nu \right)}_r\right\}}^{\beta}\;{p}^r}{{\left(r\;!\right)}^{\alpha 1}}\;\frac{{}_1S_{\alpha 1}^{\beta}\left(\nu +r,r+1,p\right)}{{}_1S_{\alpha 1}^{\beta}\left(\nu, 1,p\right)}\\ {}=\frac{{\left\{{\left(\nu \right)}_r\right\}}^{\beta}\;{p}^r}{{\left(r\;!\right)}^{\alpha 1}}\frac{{}_{\beta }F_{\alpha 1}\left(\nu +r; \kern0.24em r+1,\;r+1,\cdots,\;r+1;\kern0.24em p\right)}{{}_{\beta }F_{\alpha 1}\left(\nu; \kern0.24em 1,\;1,\cdots,\;1;\kern0.24em p\right)},\end{array} $$
where the second expression in terms of hypergeometric function is for the case when α, β are both positive integers.
Since the ECOMP (v, p, α, β) distribution is a member of exponential family (see Section 2.4), the mean is given by differentiating the logarithm of the normalizing constant with respect to p. Hence an asymptotic approximation for the mean is obtained by differentiating the logarithm of the function (9) as
$$ {p}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$\left(\alpha \beta \right)$}\right.}+\frac{1\alpha +\left(2\nu 1\right)\beta }{2\left(\alpha \beta \right)}. $$
(12)
This function approximates the mean of the ECOMP (v, p, α, β) distribution for large p and small α − β, where it is difficult to compute the approximation by truncation. A numerical illustration of this asymptotic approximation is presented in the Appendix A.2.