EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6961
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Chaundy–Bullard Type Identity and Its q-Analogue
Wathek Chammam1,∗, Mongia Khlifi2,3, Muhammad Gulistan4

1 Department of Mathematics, College of Science, Majmaah University, Al Majmaah, 11952,
Saudi Arabia
2 Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, Sfax, Tunisia
3 Research Laboratory Mathematics and Applications LR17ES11, Gabes University, Erriadh
City, 6072 Zrig, Gabes, Tunisia
4 Department of Electrical and Computer Engineering, University of Alberta, Canada

Abstract. In this paper, we use the Chaundy–Bullard combinatorial identity to prove some identi-
ties involving the Pochhammer k–symbol. In fact, these contributions generalize the results given
in the paper [O. Kouba, A Chaundy-Bullard type identity involving the Pochhammer symbol,
Indagationes Mathematicae, 34 (1), 186–198, 2023. We also present some Chaundy–Bullard type
identities satisfied by the generalized hypergeometric series.
2020 Mathematics Subject Classifications: 05A10, 05A30, 33C15, 33C05, 33C20, 33C90
Key Words and Phrases: Combinatorial identity, Chaundy-Bullard identity, Pochhammer
k–symbol, q-analogues, Gamma function, Beta function, hypergeometric

1. Introduction

Diaz and Pariguan introduced the Pochhammer k-symbol [1, p. 180], by

(x)n,k =

n−1∏
j=0

(x+ jk), n, k > 0.

When k = 1, the quantity (x)n,1 = (x)n is also called the n-th rising factorial of x.
The q-analogues of the Pochhammer k-symbol (x)n,k are given by (see [2])

[x]q;n,k =
n−1∏
j=0

[x+ jk]q, n, k > 0, (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6961

Email addresses: w.chammam@mu.edu.sa (W. Chammam),
Mongia.Khlifi@issatkas.u-kairouan.tn (M. Khlifi), mgulista@ualberta.ca (M. Gulistan)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



W. Chammam, M. Khlifi, M. Gulistan / Eur. J. Pure Appl. Math, 18 (4) (2025), 6961 2 of 9

where
[x]q =

1− qx

1− q
, x ∈ C (2)

and
lim
q→1

[x]q = x. (3)

In particular if k = 1, we obtain

[x]q;n,1 = [x]q;n =
(qx; q)n
(1− q)n

, (4)

where the symbol (x; q)n is the quantum factorial symbol defined by (see [3, 4])

(x; q)0 = 1 and (x; q)n =
n−1∏
k=0

(
1− xqk

)
, (5)

for n ≥ 1. It is easy to see that

lim
q→1

[x]q;n,k = (x)n,k and lim
q→1

[x]q;n = (x)n.

For x = n ∈ N = {1, 2, . . . } in (2), we have

[n]q =
1− qn

1− q
=

n−1∑
j=0

qj .

The q-analogue of the factorial n! is defined by (see [5])

[n]q! =


n∏

j=1

[j]q, n ≥ 1,

1, n = 0.

Moreover, the relation between the Pochhammer symbol (x)n and the classical Euler
gamma function Γ(z) is

(x)n =
Γ(x+ n)

Γ(x)
,

where
Γ(x) =

∫ ∞

0
tx−1e−tdt, x > 0.

The beta function defined by

B(x, y) =

∫ 1

0
tx−1(1− t)y−1dt =

∫ ∞

0

tx−1

(1 + t)x+y
dt, (6)

for x, y > 0.
It is clear that

B(x, y) =
Γ(x)Γ(y)

Γ(x+ y)
. (7)



W. Chammam, M. Khlifi, M. Gulistan / Eur. J. Pure Appl. Math, 18 (4) (2025), 6961 3 of 9

In [6], Jackson defined the q-analogue of the gamma function Γ(z) as

Γq(x) =
(q; q)∞
(qx; q)∞

(1− q)1−x, |q| < 1;
(q−1; q−1)∞
(q−x; q−1)∞

(q − 1)1−xq
(x
2

)
, |q| > 1,

where

(x; q)∞ =

∞∏
k=0

(
1− xqk

)
.

Also, Jackson defined the q-analogue of the beta function defined by

Bq(x, y) =

∫ 1

0
tx−1(1− qt)y−1

q dqt, x, y > 0, (8)

the relation between the q-analogue of the gamma function and the q-analogue of the beta
function is:

Bq(x, y) =
Γq(x)Γq(y)

Γq(x+ y)
. (9)

The q-binomial coefficients or the Gaussian polynomials are given by[
n

k

]
q

=
(q; q)n

(q; q)k(q; q)n−k
=

[n]q!

[k]q![n− k]q!

for 0 ≤ k ≤ n and |q| < 1. It is not difficult to prove that

lim
q→1

[
n

k

]
q

=

(
n

k

)
.

For n and m nonnegative integers, the Chaundy–Bullard identity [7–9] defined by

(1−X)n+1
m∑
k=0

(
n+ k

k

)
Xk +Xm+1

n∑
k=0

(
m+ k

k

)
(1−X)k = 1. (10)

The q-analogue of Chaundy–Bullard identity (10) defined in [10] by the equality:

m∑
k=0

[
n+ k

k

]
q

Xk
n∏

j=0

(
1−Xqj

)
+

n∑
k=0

[
m+ k

k

]
q

qkXm+1
k−1∏
j=0

(
1−Xqj

)
= 1. (11)

2. New results for the Chaundy–Bullard type identity involving the
Pochhammer p–symbol

Theorem 1. For n,m nonnegative integers and p ∈ N∗, then

(Y )n+1,p

m∑
k=0

(
n+ k

k

)
(X)k,p

(X + Y )n+k+1,p
+ (X)m+1,p

n∑
k=0

(
m+ k

k

)
(Y )k,p

(X + Y )m+k+1,p
= 1.

(12)



W. Chammam, M. Khlifi, M. Gulistan / Eur. J. Pure Appl. Math, 18 (4) (2025), 6961 4 of 9

Proof. By the identity (10), we have

(1−X)n+1
m∑
k=0

(
n+ k

k

)
Xk +Xm+1

n∑
k=0

(
m+ k

k

)
(1−X)k = 1

then for α, β > 0 and p ∈ N∗, we obtain
m∑
k=0

(
n+ k

k

)
X

α
p
+k−1

(1−X)
β
p
+n

+

n∑
k=0

(
m+ k

k

)
X

α
p
+m

(1−X)
β
p
+k−1

= X
α
p
−1

(1−X)
β
p
−1

Integrating on [0, 1] we conclude that for α, β > 0 and p ∈ N∗, we have

m∑
k=0

(
n+ k

k

)
B
(α
p
+ k,

β

p
+ n+ 1

)
+

n∑
k=0

(
m+ k

k

)
B
(β
p
+ k,

α

p
+m+ 1

)
= B

(α
p
,
β

p
,
)

hence

B
(α
p
,
β

p

)
=

m∑
k=0

(
n+ k

k

)
Γ(αp + k)Γ(βp + n+ 1)

Γ(αp + β
p + n+ k + 1)

+

n∑
k=0

(
m+ k

k

)
Γ(βp + k)Γ(αp +m+ 1)

Γ(αp + β
p +m+ k + 1)

=

m∑
k=0

(
n+ k

k

)
Γ(αp + k)

Γ(αp )

Γ(βp + n+ 1)

Γ(βp )

Γ(αp + β
p )

Γ(αp + β
p + n+ k + 1)

Γ(αp )Γ(
β
p )

Γ(αp + β
p )

+
n∑

k=0

(
m+ k

k

)
Γ(βp + k)

Γ(βp )

Γ(αp +m+ 1)

Γ(αp )

Γ(αp + β
p )

Γ(αp + β
p +m+ k + 1)

Γ(αp )Γ(
β
p )

Γ(αp + β
p )

=

m∑
k=0

(
n+ k

k

)
Γ(αp + k)

Γ(αp )

Γ(βp + n+ 1)

Γ(βp )

Γ(αp + β
p )

Γ(αp + β
p + n+ k + 1)

B
(α
p
,
β

p

)
+

n∑
k=0

(
m+ k

k

)
Γ(βp + k)

Γ(βp )

Γ(αp +m+ 1)

Γ(αp )

Γ(αp + β
p )

Γ(αp + β
p +m+ k + 1)

B
(α
p
,
β

p

)
=

m∑
k=0

(
n+ k

k

)
(αp )k(

β
p )n+1

(α+β
p )n+k+1

B
(α
p
,
β

p

)
+

n∑
k=0

(
m+ k

k

)
(βp )k(

α
p )m+1

(α+β
p )m+k+1

B
(α
p
,
β

p

)
=

m∑
k=0

(
n+ k

k

)
pk(αp )kp

n+1(βp )n+1

pn+k+1(α+β
p )n+k+1

B
(α
p
,
β

p

)
+

n∑
k=0

(
m+ k

k

)
pk(βp )kp

m+1(αp )m+1

pm+k+1(α+β
p )m+k+1

B
(α
p
,
β

p

)
=

m∑
k=0

(
n+ k

k

)
(α)k,p(β)n+1,p

(α+ β)n+k+1,p
B
(α
p
,
β

p

)
+

n∑
k=0

(
m+ k

k

)
(β)k,p(α)m+1,p

(α+ β)m+k+1,p
B
(α
p
,
β

p

)
then

(β)n+1,p

m∑
k=0

(
n+ k

k

)
(α)k,p

(α+ β)n+k+1,p
+ (α)m+1,p

n∑
k=0

(
m+ k

k

)
(β)k,p

(α+ β)m+k+1,p
= 1

The required proof is complete.



W. Chammam, M. Khlifi, M. Gulistan / Eur. J. Pure Appl. Math, 18 (4) (2025), 6961 5 of 9

Example 1. For n and m nonnegative integers and p = 1 in (12), we find the known
results [11]

(Y )n+1

m∑
k=0

(
n+ k

k

)
(X)k

(X + Y )n+k+1
+ (X)m+1

n∑
k=0

(
m+ k

k

)
(Y )k

(X + Y )m+k+1
= 1. (13)

Example 2. For n and m nonnegative integers and X = 6, Y = 4 and p = 2 in (12), we
obtain

m∑
k=0

(n+ 2)(n+ 1)(k + 2)(k + 1)

(n+ k + 1)5
+

n∑
k=0

(m+ 3)(m+ 2)(m+ 1)(k + 1)

(m+ k + 1)5
=

1

12
. (14)

Example 3. For n and m nonnegative integers and X = Y = p ∈ N∗ in (12), we have
m∑
k=0

n+ 1

(n+ k + 2)(n+ k + 1)
+

n∑
k=0

m+ 1

(m+ k + 2)(m+ k + 1)
= 1. (15)

Remark 1. For (X,Y ) = (λX, λ(1−X)) in (12) and then taking the limit as λ tends to
infinity we obtain the original Chaundy–Bullard identity (10).

3. Identity of Chaundy–Bullard type involving generalized
hypergeometric series

The generalized hypergeometric series [12], defined for complex numbers ai ∈ C and
bi ∈ C \ {0,−1,−2, ...}, for positive integers r, s ∈ N by

rFs

[
a1, . . . , ar
b1, . . . , bs

; z

]
=

∞∑
n=0

(a1)n...(ar)n
(b1)n...(bs)n

zn

n!
. (16)

The generalized basic hypergeometric series

rφs

[
a1, . . . , ar
b1, . . . , bs

; q, z

]
=

∞∑
n=0

(a1, . . . , ar; q)n
(q; q)n(b1, . . . , bs; q)n

[
(−1)nq

(n
2

)]1+s−r
zn (17)

is defined in [12, p. 125] for b1, ..., bs 6= q−m, m ∈ N,
where

(a1, a2, . . . , am; q)n = (a1; q)n(a2; q)n · · · (am; q)n. (18)

The generalized basic hypergeometric series rφs is a q-analogues of the generalized hyper-
geometric series (16).
For finite sums of generalized hypergeometric series and generalized basic hypergeometric
series, we will use the following symbols

rFs

[
a1, . . . , ar
b1, . . . , bs

; z

]
n

=
n∑

k=0

(a1)k · · · (ar)k
(b1)k · · · (bs)k

zk

k!
,



W. Chammam, M. Khlifi, M. Gulistan / Eur. J. Pure Appl. Math, 18 (4) (2025), 6961 6 of 9

and

r+1φr

[
a1, . . . , ar+1

b1, . . . , br
; q, z

]
n

=
n∑

n=0

(a1, . . . , ar+1; q)n
(q; q)n(b1, . . . , br; q)n

zn. (19)

Theorem 2. For n,m nonnegative integers and p ∈ N∗, we obtain the following equality:

(1−X)n+1
1F0

[
n+ 1
− ;X

]
m

+Xm+1
1F0

[
m+ 1
− ; 1−X

]
n

= 1. (20)

Proof. For n,m nonnegative integers, we have

1 = (1−X)n+1
m∑
k=0

(
n+ k

k

)
Xk +Xm+1

n∑
k=0

(
m+ k

k

)
(1−X)k

= (1−X)n+1
m∑
k=0

Γ(n+ 1 + k)

Γ(n+ 1)Γ(k + 1)
Xk +Xm+1

n∑
k=0

Γ(m+ 1 + k)

Γ(m+ 1)Γ(k + 1)
(1−X)k

= (1−X)n+1
m∑
k=0

(n+ 1)k
Xk

k!
+Xm+1

n∑
k=0

(m+ 1)k
(1−X)k

k!

= (1−X)n+1
1F0

[
n+ 1
− ;X

]
m

+Xm+1
1F0

[
m+ 1
− ; 1−X

]
n

.

The required proof is complete.

Theorem 3. For n,m nonnegative integers, we have

n∏
j=0

(
1−Xqj

)
1φ0

[
qn+1

− ; q,X

]
m

+Xm+1
2φ1

[
qm+1, X

0
; q, q

]
n

= 1. (21)

Proof. For n,m nonnegative integers, we have

(a; q)n+m = (a; q)n(aq
n; q)m

then, by (11) we obtain

1 =
m∑
k=0

[
n+ k

k

]
q

Xk
n∏

j=0

(
1−Xqj

)
+

n∑
k=0

[
m+ k

k

]
q

qkXm+1
k−1∏
j=0

(
1−Xqj

)
=

n∏
j=0

(
1−Xqj

) m∑
k=0

(q; q)n+k

(q; q)n(q; q)k
Xk +Xm+1

n∑
k=0

(q; q)m+k(X; q)k
(q; q)m(q; q)k

qk

=

n∏
j=0

(
1−Xqj

) m∑
k=0

(q; q)n(q
n+1; q)k

(q; q)n(q; q)k
Xk +Xm+1

n∑
k=0

(q; q)m(qm+1; q)k(X; q)k
(q; q)m(q; q)k

qk



W. Chammam, M. Khlifi, M. Gulistan / Eur. J. Pure Appl. Math, 18 (4) (2025), 6961 7 of 9

=

n∏
j=0

(
1−Xqj

) m∑
k=0

(qn+1; q)k
Xk

(q; q)k
+Xm+1

n∑
k=0

(qm+1; q)k(X; q)k
qk

(q; q)k

=
n∏

j=0

(
1−Xqj

) m∑
k=0

(qn+1; q)k
Xk

(q; q)k
+Xm+1

n∑
k=0

(qm+1; q)k(X; q)k
(0; q)k

qk

(q; q)k

=
n∏

j=0

(
1−Xqj

)
1φ0

[
qn+1

− ; q,X

]
m

+Xm+1
2φ1

[
qm+1, X

0
; q, q

]
n

.

We find the result.

Now, we are interested in relations between the identity of Chaundy–Bullard involving
the Pochhammer p–symbol and hypergeometric series asserted in the following Theorem.

Theorem 4. For n,m nonnegative integers and p ∈ N∗, we obtain the following equality:

(Y )n+1,p

(X + Y )n+1,p
2F1

[
X
p , n+ 1

X+Y
p + n+ 1

; 1

]
m

+
(X)m+1,p

(X + Y )m+1,p
2F1

[
Y
p ,m+ 1

X+Y
p +m+ 1

; 1

]
n

= 1.

(22)

Proof. For n,m nonnegative integers and p ∈ N∗, we have

(Y )n+1,p

m∑
k=0

(
n+ k

k

)
(X)k,p

(X + Y )n+k+1,p

= (Y )n+1,p

m∑
k=0

Γ(n+ k + 1)

Γ(n+ 1)Γ(k + 1)

pk(Xp )k

pn+k+1(X+Y
p )n+k+1

= (Y )n+1,p

m∑
k=0

(X
r

)
k

Γ(n+ k + 1)

Γ(n+ 1)Γ(k + 1)

Γ(X+Y
p )

pn+1Γ(X+Y
p + n+ k + 1)

Γ(X+Y
p + n+ 1)

Γ(X+Y
p + n+ 1)

= (Y )n+1,p

m∑
k=0

(X
p

)
k

Γ(n+ 1 + k)

Γ(n+ 1)

Γ(X+Y
p + n+ 1)

Γ(X+Y
p + n+ 1 + k)

Γ(X+Y
p )

pn+1Γ(X+Y
p + n+ 1)

1

Γ(k + 1)

= (Y )n+1,p

m∑
k=0

(Xp )k(n+ 1)k

(X+Y
p + n+ 1)kpn+1(X+Y

p )n+1

1

Γ(k + 1)

=
(Y )n+1,p

(X + Y )n+1,p

m∑
k=0

(Xp )k(n+ 1)k

(X+Y
p + n+ 1)k

1

k!

=
(Y )n+1,p

(X + Y )n+1,p
2F1

[
X
p , n+ 1

X+Y
p + n+ 1

; 1

]
m

.

Then

(Y )n+1,p

m∑
k=0

(
n+ k

k

)
(X)k,p

(X + Y )n+k+1,p
=

(Y )n+1,p

(X + Y )n+1,p
2F1

[
X
p , n+ 1

X+Y
p + n+ 1

; 1

]
m



W. Chammam, M. Khlifi, M. Gulistan / Eur. J. Pure Appl. Math, 18 (4) (2025), 6961 8 of 9

Consequently, we have the equality

(X)m+1,p

n∑
k=0

(
m+ k

k

)
(Y )k,p

(X + Y )m+k+1,p
=

(X)m+1,p

(X + Y )m+1,p
2F1

[
Y
p ,m+ 1

X+Y
p +m+ 1

; 1

]
n

We using (12), we obtain the result.

Corollary 1. For n,m nonnegative integers, we have

(Y )n+1

(X + Y )n+1
2F1

[
X,n+ 1

X + Y + n+ 1
; 1

]
m

+
(X)m+1

(X + Y )m+1
2F1

[
Y,m+ 1

X + Y +m+ 1
; 1

]
n

= 1.

(23)

Proof. For n,m nonnegative integers and p = 1 in (22), we obtain the result.

Example 4. If X = Y , m = n in (22) we have

2F1

[
X
p , n+ 1

2X
p + n+ 1

; 1

]
n

=
(2X)n+1,p

2(X)n+1,p
. (24)

Example 5. If X = Y , m = n and p = 1 in (22) we obtain

2F1

[
X,n+ 1

2X + n+ 1
; 1

]
n

=
(2X)n+1

2(X)n+1
. (25)

4. Conclusion and Perspectives

In this paper, we have established a generalization of the classical Chaundy–Bullard
identity together with its q-analogue. Our approach, based on combinatorial manipulations
of generalized factorials and hypergeometric-type series, highlights the structural links
between binomial identities, q-series, and special functions. Several illustrative examples
were provided, showing how known formulas (such as the Beta integral and its q-extension)
can be recovered as particular cases of our results.

Beyond the intrinsic combinatorial interest of such identities, these results open several
directions for future research:

• exploring further extensions involving multiple parameters, higher-order factorials
or multivariate generalizations;

• investigating connections with orthogonal polynomials, especially those arising in
the Askey scheme and their q-analogues;

• applying these identities to the study of partition functions, q-series transformations,
and related problems in analytic number theory;



W. Chammam, M. Khlifi, M. Gulistan / Eur. J. Pure Appl. Math, 18 (4) (2025), 6961 9 of 9

• examining possible applications in approximation theory, where Beta-type integrals
and their discrete versions naturally arise.

We believe that the framework introduced here provides a unifying point of view for
various classical and modern identities, and may stimulate further developments at the
intersection of combinatorics, special functions and q-series.

Acknowledgements

The authors extends the appreciation to the Deanship of Postgraduate Studies and
Scientific Research at Majmaah University for funding this research work through the
project number(ICR-2025-2029).

References

[1] R. Díaz and E. Pariguan. On hypergeometric functions and pochhammer k-symbol.
Divulg. Mat, 15(2):179–192, 2007.

[2] M. Khlifi, W. Chammam, and B N Guo. Several identities and relations related to
q–analogues of pochhammer k-symbol with applications to fuss-catalan–qi numbers.
Afr. Mat., 35:21, 1905.

[3] K. Brahim and H. Elmonser. Some new q-versions of ramanujan’s master theorem.
Complex Anal. Oper. Theory, 17:17, 2023.

[4] H. Elmonser. Symmetric q-extension of lambda-apostol-euler polynomials via umbral
calculus. Indian. J Pure Appl Math, 54:583–594, 2023.

[5] W. Chammam. Several formulas and identities related to catalan–qi and q-catalan–qi
numbers. Indian. J Pure Appl Math, 50:1039–1048, 2019.

[6] F. H. Jackson. The basic gamma-function and the elliptic functions. R. Soc. Lond.
Proc. Ser. A Math. Phys. Eng. Sci., 76(508):127–144, 1905.

[7] T. W. Chaundy and J. E. Bullard. John smith’s problem. Math. Gaz, 44:253–260,
1960.

[8] T. H. Koornwinder and M. J. Schlosser. On an identity of chaundy and bullard. i.
Indag. Math. (N.S.), 19(2):239–261, 2008.

[9] T. H. Koornwinder and M. J. Schlosser. On an identity of chaundy and bullard.
ii.more history. Indag. Math. (N.S.), 24(1):174–180, 2013.

[10] V. J. W. Guo and S. D. Wang. A symmetric generalization of an identity of andrews
and yee. Discrete Mathematics, 342(7):2112–2115, 2019.

[11] O. Kouba. A chaundy-bullard type identity involving the pochhammer symbol. Inda-
gationes Mathematicae, 34(1):186–189, 1905.

[12] N. M. Temme. Special Functions: An Introduction to Classical Functions of Mathe-
matical Physics. A Wiley-Interscience Publication, John Wiley Sons, Inc., New York,
1996.