1. Introduction and Preliminaries
Kumar and Consul [1] develop a recursive relation upon the negative moments of power series distribution. The recurrence relation for the negative moments of the Poisson distribution was first derived by Chao and Strawderman [2], after which it is shown by Kumar and Consul [1] as a special case of their result. Using the recurrence relation for the negative moments of the Lagrangian binomial distribution, Kumar and Consul [1] have established the binomial and negative binomial distributions.

Ahmad and Saboor [3] proved many properties of the hypergeometric series. Besides, the following series have been provided by Ahmad and Saboor [3] and (1)Fqpa1,k,a2,k,…,ap,k;b1,k,b2,k,…,bq,k;z=1+a1k·a2k⋯apkb1k·b2k⋯bqkz+a1a1+1ka2a2+1k⋯apap+1kb1b1+1kb2b2+1k⋯bqbq+1kz22!+⋯is generalized hypergeometric series, where (λik)n=(λik)(λi+1)k(λi+2)k⋯(λi+n-1)k, (λik)0=1, i=1,2,…,p, k=1,2,…. If k=1, then (2)Fqp=∑n=0∞a1na2n⋯apnb1nb2n⋯bqnznn!is hypergeometric series.

In this paper, the recurrence relation for negative moments along with negative factorial moments of some discrete distributions can be obtained. These relations have been derived with properties of the hypergeometric series.

In the next part, some necessary definitions have been introduced.

Let X be a generalized negative binomially distributed random variable with parameters θ, β and the probability mass function is (3)Pxθ,β=mm+βxm+βxxθx1-θm+βx-x,x=0,1,2,…,where 0<θ<1, |θβ|<1, 1≤β≤θ-1, and m>0 and a constant.

Let X be a generalized Poisson distribution with parameters λ, θ and the probability mass function is (4)Pxθ=e-λ1+θxλx1+θxx-1x!,x=0,1,…,where 0<λ<∞, |λθ|<1, and 0<θ<1.

Let the random variable X be equipped with a generalized poisson-negative-binomial distribution with parameters λ, θ, and β; the probability mass function is (5)Pxθ,β=e-λ1+θs∑s=0∞λs1+θss-1s!mm+βxm+βxxθx1-θm+βx-x,x=0,1,2,…,where 0<θ<1, 0<λ<∞, |λθ|<1, |θβ|<1, 1≤β≤θ-1, s≥0, s∈Z, and m>0 and a constant.

Let X be a generalized logarithmic distributed random variable with parameters θ and β; the probability mass function is (6)Pxθ,β=1βxβxxθx1-θβx-x-log1-θ,x=1,2,…,where 0<θ<1, 0<θβ<1, and β≥1.

2. The Recurrence Relation for Inverse Moments of Some Discrete Distributions
In this section, some recurrence relations for inverse moments of some discrete distributions can be obtained with the properties of the generalized hypergeometric series functions.

Theorem 1.
Let X be a generalized negative binomial random variable with parameters θ, β, for 0<θ<1, |θβ|<1, 1≤β≤θ-1, and probability mass function is defined in (3), and then the inverse moment of first order is given by (7)EX+A-1=mθ1-θm+β-1A+1F32A+1,m+n+1β-n,1;A+2,2;θ1-θ1-β,where A≥0, n=0,1,2,…, and m>0 and a constant.

Proof.
Since X is a generalized negative binomial random variable with parameters θ, β, then (8)EX+A-1=∑x=1∞mm+βxm+βxxθx1-θm+βx-x1x+A=mθ1-θm+β-1A+11+A+1m+2β-1A+2·2!θ1-θ1-β+A+1m+3β-1m+3β-2A+3·3!θ21-θ21-β+⋯=mθ1-θm+β-1A+11+A+1m+2β-1·1A+2·2·1!θ1-θ1-β+A+1A+2m+3β-1m+3β-2·1·2A+2A+3·2·3·2!θ1-θ1-β2+⋯=mθ1-θm+β-1A+1F32A+1,m+n+1β-n,1;A+2,2;θ1-θ1-β.

Theorem 2.
Let X be a generalized negative binomially distributed random variable with parameters θ and β, and probability mass function is defined in (3). Then the following relation holds: (9)Azp-A-1EX+A-1=A1-AEX+A-1-1-P11-z-p+1+Q0,where Q0=[A+(p-A-1)z]F21[1,p;2;z], p=m+(n+1)β-n, n=0,1,2,…, z=θ/(1-θ)1-β, and P1=P(X=1)=mθ(1-θ)m+β+1.

Proof.
From Theorem 1, we have (10)EX+A-1=P1A+1F32A+1,p,1;A+2,2;z,using the identity (see [4, page 85]) (11)1-xF32α1,α2,α3;β1,β2;x=F32α1-1,α2,α3;β1,β2;x+x·α2-β1α3-β1β1β2-β1F32α1,α2,α3;β1+1,β2;x+xα2-β2α3-β2β2β1-β2·F32α1,α2,α3;β1,β2+1;x.For α1=A+1, α2=p, α3=1, β1=2, β2=A+1, and x=z we have (12)1-zF32A+1,p,1;A+1,2;z=F32A,p,1;A+1,2;z+2-pz2A-1F32A+1,p,1;A+1,3;z+Azp-A-1A2-1F32A+1,p,1;A+2,2;z,and we know that (13)F32A+1,p,1;A+1,2;z=F211,p;2;z,F32A+1,p,1;A+1,3;z=F211,p;3;z.Then (12) becomes (14)1-zF211,p;2;z=F32A,p,1;A+1,2;z+2-pz2A-1F211,p;3;z+Azp-A-1A2-1F32A+1,p,1;A+2,2;z.Using another identity (see [4, page 71]) (15)a+b-czF21a,b;c;z=a1-zF21a+1,b;c;z-c-ac-bzcF21a,b;c+1;z,and for a=1, b=p, c=2, and z=z we have (16)1+p-2zF211,p;2;z=1-zF212,p;2;z-2-pz2F211,p;3;zas (17)F212,p;2;z=F10p;-;z=1-z-p.From (14) and (16) we have (18)1-zF211,p;2;z=F32A,p,1;A+1,2;z+Azp-A-1A2-1F32A+1,p,1;A+2,2;z-1+p-2zA-1F211,p;2;z+1-z-p+1A-1.Rearranging we get (19)A+p-A-1zA-1F211,p;2;z-1-z-p+1A-1=AP1EX+A-1-1+Azp-A-1P1A-1EX+A-1.By collating, we get the result (9).

Theorem 3.
Let the random variable X be equipped with a generalized Poisson distribution with parameters θ and λ, and probability mass function is defined in (4). Then (20)AzEX+A-1=A1-AEX+A-1-1+AP1F111;2;z-P1,and P1=P(X=1)=λe-λ(1+θ), z=λ(1+(n+1)θ)/eλθ, and n=0,1,2,….

Proof.
Using the identity (see [4, page 84]) (21)F22α1,α2;β1,β2;x=F22α1-1,α2;β1,β2;x+xα2-β1β1β2-β1F22α1,α2;β1+1,β2;x+xα2-β2β2β1-β2F22α1,α2;β1,β2+1;x.By the same as Theorem 2 calculating, we get the result (20).

Theorem 4.
Let the random variable X be equipped with a generalized Poisson-negative-binomial distribution with parameters λ, θ, and β, and probability mass function is defined in (5); then (22)Azp-A-1EX+A-1=A1-AEX+A-1-1-P11-z-p+1+Q1,where P1=P(X=1)=e-λ1+θs∑s=0∞(λs(1+θs)s-1/s!)mθ(1-θ)m+β-1, p=m+(n+1)β-n, n=0,1,2,…, Q1=P1[A+(p-A-1)z]F21[1;p;2;z], and z=θ/(1-θ)1-β.

Proof.
Using identities (11) and (15), by the same as Theorem 2 calculating, we get the result (22).

Theorem 5.
Let X be a generalized logarithmic distributed random variable with parameters θ and β, and probability mass function is defined in (6). Then the following recurrence relation holds: (23)Azp-A-1EX+A-1=A1-AEX+A-1-1-P11-z-p+1+Q2,where P1=P(X=1)=θ(1-θ)β-1/[-log(1-θ)], Q2=P1[A+(p-A-1)z]F21[1,p;2;z], p=(n+1)β-n, n=0,1,2,…, and z=θ/(1-θ)1-β.

Proof.
Using identities (11) and (15), by the same as Theorem 2 calculating, we get the result (23).

3. Recurrence Relation for Inverse Factorial Moments of Discrete Distributions
In this section, some recurrence relations for inverse factorial moments of some discrete distributions can be obtained with the properties of the generalized hypergeometric series functions.

Theorem 6.
Let X be a generalized negative binomial random variable with parameters θ, β, for 0<θ<1, |θβ|<1, 1≤β≤θ-1, and probability mass function is defined in (3), and then the factorial inverse moment of first order is given by (24)E∏i=1lx+i-1=mθ1-θm+β-1l+1!F322,m+n+1β-n,1;l+2,2;θ1-θ1-β,where l≥1 and l∈Z+, n=0,1,2,….

Proof.
Since X is a generalized negative binomial random variable with parameters θ, β, then (25)E∏i=1lx+i-1=∑x=1lmm+βxm+βxxθx1-θm+βx-x∏i=1l1x+i=∑x=1lmm+βxm+βxxθx1-θm+βx-xx!l+x!=mθ1-θm+β-1l+1!1+2m+2β-1·1l+2·2·1!θ1-θ1-β+2·3m+3β-1m+3β-2·1·2l+2l+3·2·3·2!θ1-θ1-β2+⋯=mθ1-θm+β-1l+1!F21m+n+1β-n,1;l+2;θ1-θ1-β.

Theorem 7.
Let X equipped with generalized negative binomial probability distribution be defined in (3) with parameters θ and β; suppose μ-[l]′ is the lth negative factorial moment of X. Then the relation (26)l+1p-l-2zμ-l+1′=p-2l-2z+l+1μ-l′-1-zμ-l-1′holds for l=2,3,4,…, and p=m+(n+1)β-n, n=0,1,2,…, z=θ/(1-θ)1-β.

Proof.
From Theorem 6, we have (27)μ-l′=mθ1-θm+β-1l+1!F211,p;l+2;z.Using the identity (see [4, page 71]) (28)a-c+1F21a,b;c;z=aF21a+1,b;c;z-c-1F21a,b;c-1;z,and for a=1, b=p, c=l+3, and z=z we have (29)F211,p;l+3;z=l+2l+1F211,p;l+2;z-1l+1F212,p;l+3;z.Using the identity (see [4, page 71]) (30)1-zF21a1,b1;c1;z=F21a1-1,b1;c1;z-c1-b1c1zF21a1,b1;c1+1;z,for a1=2, b1=p, c=l+2, and z=z; rearranging we have (31)F212,p;l+3;z=l+21-zp-l-2zF212,p;l+2;z-l+2p-l-2zF211,p;l+2;z.Substituting in (29), we get (32)F211,p;l+3;z=l+2p-l-2z+1l+1p-l-2zF211,p;l+2;z-l+21-zl+1p-l-2zF212,p;l+2;z.Consider F21[2,p;l+2;z] and using (28), for a=1, b=p, c=l+2, and z=z we have (33)F212,p;l+2;z=l+1F211,p;l+1;z-lF211,p;l+2;z.Substituting in (32), we get (34)F211,p;l+3;z=l+2p-l-2z+1l+1p-l-2zF211,p;l+2;z-l+21-zp-l-2zF211,p;l+1;z+ll+21-zl+1p-l-2zF211,p;l+2;z=l+2p-2l-2z+l+1l+1p-l-2zF211,p;l+2;z-l+21-zp-l-2zF211,p;l+1;z.Rearranging we have (35)l+2!mθ1-θm+β-1μ-l+1′=l+2p-2l-2z+l+1l+1p-l-2zl+1!mθ1-θm+β-1μ-l′-l+21-zl+1p-l-2zl+1!mθ1-θm+β-1μ-l-1′,and by collating, we get the result (26).

Theorem 8.
Let the random variable X equipped with a generalized Poisson distribution be defined in (4) with parameters λ and θ, and μ-[l]′ is the lth negative factorial moment of X. Then the relation (36)l+1zμ-l+1′=l+z+1μ-l′-μ-l-1′holds for l=2,3,4,…, z=(1+(n+1)θ)λ/eλθ, λ,θ>0, n=0,1,2,….

Proof.
Using the identity (see [4, pages 82 and 84]) (37)α1-β1+1F22α1,α2;β1,β2;x=α1F22α1+1,α2;β1,β2;x-β1-1F22α1,α2;β1-1,β2;x,F22α1,α2;β1,β2;x=F22α1-1,α2;β1,β2;x+xα2-β1β1β2-β1F22α1,α2;β1+1,β2;x+xα2-β2β2β1-β2F22α1,α2;β1,β2+1;x,by the same as Theorem 7 calculating, we get the result (36).

Theorem 9.
Let X equipped with generalized logarithmic series distribution be defined in (6) with parameters θ and β, and μ-[l]′ is the lth inverse factorial moment of X. Then the relation (38)l+1p-l-2zμ-l+1′=p-2l-2z+l+1μ-l′-1-zμ-l-1′,holds for l=2,3,4,…, p=(n+1)β-n, n=0,1,2,…, z=θ/(1-θ)1-β.

Proof.
Using the identities (28) and (30), by the same as Theorem 7 calculating, we get the result (38).

4. The Accurate Value for Inverse Moments of Some Discrete Distributions
In this section, some accurate values for inverse moments of some discrete distributions can be obtained with their recurrence relations.

Theorem 10.
Let X be a generalized negative binomially distributed random variable with parameters θ and β, and probability mass function is defined in (3). Then the accurate value for inverse moment of first order is given by (39)AEX+A-1=∑i=-1A-3-1iA+1-ii·P1p-A-1i+1·zi+11-z-p+1+-1A-13A-2·P1p-A-1A-1·zA-1F211,p;3;z+∑i=0A-2-1i·A+1-iiA-i+p-A+1-izp-A-1i+1·zi+1F211,p;2;z,where p=m+(n+1)β-n, n=0,1,2,…, z=θ/(1-θ)1-β, P1=P(X=1)=mθ(1-θ)m+β+1.

Proof.
From (9), we have (40)Azp-A-1EX+A-1=A1-AEX+A-1-1-P11-z-p+1+A+p-A-1zF211,p;2;z,and let A=A-1; we get (41)A-1zp-AEX+A-1-1=A-12-AEX+A-2-1-P11-z-p+1+A-1+p-AzF211,p;2;z,and hence (42)AEX+A-1=AA-1A-2p-A-1p-A·z2EX+A-2-1+A·P11-z-p+1p-A-1p-A·z2-AA-1+p-Azp-A-1p-A·z2F211,p;2;z-P11-z-p+1p-A-1·z+A+p-A-1zp-A-1·zF211,p;2;z,and so on; repeat the above steps; we can get (43)AEX+A-1=-1A-12A-1p-A-1A-1·zA-1·EX+1-1+-1A-1·3A-2·P11-z-p+1p-A-1A-1·zA-1+-1A-2·4A-3·P11-z-p+1p-A-1A-2·zA-2+⋯+-12A1·P11-z-p+1p-A-12·z2+-11·A+10·P11-z-p+1p-A-11·z+-1A-2·3A-22+p-3zp-A-1A-1·zA-1·F211,p;2;z+-1A-3·4A-33+p-4zp-A-1A-2·zA-2·F211,p;2;z+⋯+-11·A1A-1+p-Azp-A-12·z2·F211,p;2;z+-10·A+10A+p-A-1zp-A-11·z·F211,p;2;z,and, from (10), we have (44)EX+1-1=P12F211,p;3;z,so we obtain the result (39).

Theorem 11.
Let the random variable X be equipped with a generalized Poisson distribution with parameters θ and λ, and probability mass function is defined in (4), and then the accurate value for inverse moment of first order is given by (45)AEX+A-1=∑i=-1A-3-1iA-ii+1·P1zi+2+-1A-13A-2·P1zA-1F111;3;z+∑i=0A-2-1iA-ii+1·P1zi+1F111;2;z,where P1=P(X=1)=λe-λ(1+θ), z=λ(1+(n+1)θ)/eλθ, and n=0,1,2,….

The proof is the same as Theorem 10, omitted here.

Theorem 12.
Let the random variable X be equipped with a generalized poisson-negative-binomial distribution with parameters λ, θ, and β, and probability mass function is defined in (5), and then the accurate value for inverse moment of first order is given by (46)AEX+A-1=∑i=-1A-3-1iA-ii+1·P1p-A-1i+2·zi+21-z-p+1+-1A-13A-2·P1p-A-1A-1·zA-1F211,p;3;z+∑i=0A-2-1i·A+1-iiA-i-p-A+1-iz·P1p-A-1i+1·zi+1F211,p;2;z,where P1=P(X=1)=e-λ1+θs∑s=0∞(λs(1+θs)s-1/s!)mθ(1-θ)m+β-1, p=m+(n+1)β-n, n=0,1,2,…, and z=θ/(1-θ)1-β.

The proof is the same as Theorem 10, omitted here.

Theorem 13.
Let X be a generalized logarithmic distributed random variable with parameters θ and β, and probability mass function is defined in (6). Then the accurate value for inverse moment of first order is given by (47)AEX+A-1=∑i=-1A-3-1iA-ii+1·P1p-A-1i+2·zi+21-z-p+1+-1A-13A-2·P1p-A-1A-1·zA-1F211,p;3;z+∑i=0A-2-1i·A+1-iiA-i+p-A+1-iz·P1p-A-1i+1·zi+1F211,p;2;z,where P1=P(X=1)=θ(1-θ)β-1/[-log(1-θ)], p=(n+1)β-n, n=0,1,2,…, and z=θ/(1-θ)1-β.

The proof is the same as Theorem 10, omitted here.