/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 8, No. 2, 2015, 214-231
ISSN 1307-5543 – www.ejpam.com

Some Properties of the p-adic Beta Function

Hamza Menken∗, Özge Çolakoğlu

Department of Mathematics, Science and Arts Faculty, Mersin University, Mersin, Turkey

Abstract. In the present work we consider a p-adic analogue of the classical beta function by using Y.
Morita’s p-adic gamma function. We obtain some elementary properties of the p-adic beta function.
We give some relations between the classical beta and the p-adic beta functions at the values of natural
numbers.

2010 Mathematics Subject Classifications: 11S80; 11E95

Key Words and Phrases: p-adic number, p-adic gamma function, p-adic beta function

1. Introduction

Let p be fixed prime number. It is well known that the p−adic valuation of any x ∈ Q,
x 6= 0 is determined by the formula

x = pvp(x).
a

b

where vp(x) ∈ Z and ab is not divided by p. The p-adic norm |·|p is defined by

|x |p =
¨

p−vp(x), x 6= 0

0, x = 0.

By Qp we denote the completion of rational numbers field Q with respect to the p-adic norm
|·|p. The ring of p-adic integers is the valuation ring

Zp =
�

x ∈ Qp : |x |p ≤ 1
	

.

Note that every x ∈ Zp can be written in the form

x = b0 + b1p+ b2p2 + . . .+ bnpn + . . .

∗Corresponding author.

Email addresses: hmenken@mersin.edu.tr (H. Menken), ozgecolakoglu@mersin.edu.tr (Ö. Çolakoğlu)

http://www.ejpam.com 214 c© 2015 EJPAM All rights reserved.



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 215

with 0≤ bi ≤ p− 1; and also, every x ∈ Qp can be written in the form

x = b−n0
p−n0 + . . .+ b0 + b1p+ b2p2 + . . .+ bnpn + . . .=

∑

n≥−n0

bnpn

with 0≤ bi ≤ p− 1 and −n0 = vp(x) (for details see [12]).
The classical gamma function is an extension of the factorial function and is defined by the

formula

Γ(x) =

∞
∫

0

t x−1e−t d t

for all Re (x)> 0 [1]. The basic properties of the classical gamma function are following:

(i) Γ(n+ 1) = n! for all non negative integer n

(ii) Γ(z + 1) = zΓ(z) (Re (z)> 0)

(iii) Γ(1− z)Γ(z) = π
sin(πz)

(Re (z)> 0)

(iv) Γ(1
2) =
p
π.

It is well known that the classical beta function B(x , y) is defined by

B(x , y) =
Γ(x)Γ(y)

Γ(x + y)

and it has the integral representation

B(x , y) =

1
∫

0

t x−1 (1− t)y−1 d t

for all Re (x), Re
�

y
�

> 0. The basic properties of the classical beta function are the following:

(i) B(x , y) = B(y, x)

(ii) B(x + 1, y) = B(x , y) x
x+y

(iii) B(x , y + 1) = B(x , y)
y

x+y

(iv) B(x , y)B(x + y, 1− y) = π

x sin(πy)

(v)

�

n

k

�

= 1
(n+1)B(n−k+1,k+1) (n, k ∈ N, k ≤ n)

(vi) B(1
2 , 1

2) = π



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 216

(vii) B(x + 1, y) + B(x , y + 1) = B(x , y)

(viii) B(x , y + 1) = y
x B(x + 1, y) =

y
x+y B(x , y)

(ix) B(x , y)B(x + y, z)B(x + y + z, w) =
Γ(x)Γ(y)Γ(z)Γ(w)

Γ(x+y+z+w)

where Re (x) , Re
�

y
�

, Re (z) , Re (w)> 0. The p-adic analogue of the classical gamma function
depends on the p-adic version of factorial function. The p-adic version of factorial function is
defined by

(n!)p :=
∏

1≤ j ≤ n

( j, p) = 1

j

The function f (n) = (−1)n+1(n!)p can be interpolated and the p-adic gamma function Γp is
defined as follows:

Definition 1 ([10]). The p-adic gamma function Γp is the continuous extension to Zp of

n 7→ (−1)n
∏

1≤ j < n

( j, p) = 1

j(n≥ 2).

Moreover, Γp : Zp→ Qp function is defined by

Γp(x) := lim
n→x
(−1)n
∏

1≤ j < n

( j, p) = 1

j.

According the definition of p-adic factorial function we conclude that:

Corollary 1. Γp(n+ 1) = (−1)n+1(n!)p (n ∈ N).
To prove our results we use the following properties of p-adic gamma function:

Proposition 1 ([12]). Let p 6= 2. Then Γp has the following properties:

(i) For all x ∈ Zp

Γp(x + 1) = hp(x)Γp(x) (1)

where

hp(x) :=

¨

−x if |x |p = 1

−1 if |x |p < 1

(ii) Γp(0) = 1, Γp(1) = −1, Γp(2) = 1. For all x ∈ Zp we have
�

�Γp(x)
�

�

p
= 1

(iii) For all x , y ∈ Zp
�

�Γp(x)− Γp(y)
�

�

p
≤
�

�x − y
�

�

p
. (2)



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 217

Also, the properties (i) and (ii) hold for p = 2, and the instead of (iii) the relations

�

�Γ2(x)− Γ2(y)
�

�

2 ≤
�

�x − y
�

�

2 (x , y ∈ Z2,
�

�x − y
�

�

2 6=
1

4
)

�

�Γ2(x)− Γ2(y)
�

�

2 ≤ 2
�

�x − y
�

�

2 (x , y ∈ Z2,
�

�x − y
�

�

2 =
1

4
)

hold.

Proposition 2 ([12]). A formula for Γp(−n) (n ∈ N) is given by

Γp(−n) = (−1)n+1−
�

n
p

�

(Γp(n+ 1))−1. (3)

Proposition 3 ([7, 12]). If p 6= 2 then

Γp(x)Γp(1− x) = (−1)ℓ(x) (x ∈ Zp) (4)

and for p = 2
Γp(x)Γp(1− x) = (−1)σ1(x)+1 (x ∈ Z2) (5)

where ℓ : Zp →
�

1,2, . . . , p
	

assigns to x ∈ Zp its residue ∈ �1,2, . . . , p
	

modulo pZp and
where σ1 is defined by the formula

σ1(

∞
∑

j=0

a j2
j) = a1

Corollary 2 ([12]). Let p 6= 2. We get

Γp(
1

2
)2 = (−1)

ℓ( 1
2 ) (6)

Now ℓ(1
2) = ℓ(

1
2(p+ 1)) = 1

2(p+ 1) so that

Γp(
1

2
)2 =

¨

1 if p ≡ 3 (mod 4)

−1 if p ≡ 1 (mod 4)
(7)

Proposition 4 ([12]). Let n ∈ N and let sn be sum of the digits of n =
s
∑

j=0
a j p

j (as 6= 0) in base

p. Then

(i) Γp(n+ 1) = (−1)n+1 n!
�

n
p

�

!p[
n
p ]

(ii) Γp(p
n) = (−1)p pn!

pn−1!ppn−1

(iii) n!= (−1)n+1−s(−p)(n−sn)/(p−1)
n

Π
j=0
Γp

��

n
p j

�

+ 1
�



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 218

(iv) pn!= (−1)p(−p)(p
n−1)/(p−1)

n

Π
j=0
Γp

�

p j
�

.

The p-adic gamma function have been considered by many authors (see [3–10, 13]). We
note that another p-adic analogue of classical gamma function was constructed by G. Over-
holtzer [11], but we consider Morita’s p-adic gamma function.

In 1980 the p-adic beta function is used in Dwork cohomology and an cohomological in-
terpretation of p-adic beta function is given by M. Boyarsky [4]. In 2006 F. Baldassarri [2]
considered two constructions of the p-adic beta functions as the p-adic etale and p-adic crys-
talline beta functions. Also, some comparisons between the p-adic etale and p-adic crystalline
beta functions with relations via Fontaine’s periods are given.

In the present work we study a p-adic analogue of classical beta function by using Morita’s
p-adic gamma function, and we obtain some elemantary properties of the p-adic beta function.

2. Main Results

Naturally a p-adic analogue of the classical beta function can be defined as follows.

Definition 2. The p-adic beta function Bp : Zp ×Zp→ Qp is defined by the formula

Bp(x , y) :=
Γp (x)Γp(y)

Γp(x + y)
, x , y ∈ Zp. (8)

We investigate some properties of the p-adic beta function. Now, we give basic properties
of the p-adic beta function.

Theorem 1. The p-adic beta function is symmetric. Namely,

Bp(x , y) = Bp(y, x)

for x , y ∈ Zp.

Proof. From Definition 2, we can prove that the p-adic beta function is symmetric:

Bp(x , y) =
Γp (x)Γp(y)

Γp(x + y)

=
Γp

�

y
�

Γp(x)

Γp(y + x)

=Bp(y, x)

for x , y ∈ Zp.



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 219

Theorem 2. For x , y ∈ Zp, then

Bp(x , y)Bp(x + y, 1− y) =







(−1)ℓ(y)

hp(x)
, p 6= 2

(−1)σ1(y)+1

hp(x)
p = 2

where

hp(x) :=

¨

−x if |x |p = 1

−1 if |x |p < 1

and ℓ : Zp →
�

1,2, . . . , p
	

assigns to x ∈ Zp its residue ∈ �1,2, . . . , p
	

modulo pZp and σ1 is

defined by the formula

σ1(

∞
∑

j=0

a j2
j) = a1

Proof. Let p 6= 2. From Definition 2 and Proposition 1 it follows that

Bp(x , y)Bp(x + y, 1− y) =
Γp (x)Γp(y)

Γp(x + y)

Γp

�

x + y
�

Γp(1− y)

Γp(x + 1)

=
Γp (x)Γp(y)Γp(1− y)

Γp(x + 1)

=
Γp (x)Γp(y)Γp(1− y)

Γp(x)hp(x)

=
Γp(y)Γp(1− y)

hp(x)
,

and by Proposition 3 we obtain that

Bp(x , y)Bp(x + y, 1− y) =
(−1)ℓ(y)

hp(x)
.

In similar way, we can prove the theorem for p = 2.

Theorem 3. The equality

Bp(x + 1, y) =
hp(x)

hp(x + y)
Bp(x , y)

holds for all x , y ∈ Zp.

Proof. By using Definition 2 and Proposition 1 we have that

Bp(x + 1, y) =
Γp (x + 1)Γp(y)

Γp(x + 1+ y)



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 220

=
Γp (x)hp(x)Γp(y)

Γp((x + y) + 1)

=
Γp (x)hp(x)Γp(y)

Γp(x + y)hp(x + y)

=
hp(x)

hp(x + y)

Γp (x)Γp(y)

Γp(x + y)

=
hp(x)

hp(x + y)
Bp(x , y).

Theorem 4. The equality

Bp(x , y + 1) =
hp(y)

hp(x + y)
Bp(x , y)

holds for all x , y ∈ Zp.

Proof. From Definition 2 and Proposition 1 we get

Bp(x , y + 1) =
Γp (x)Γp(y + 1)

Γp(x + y + 1)

=
Γp (x)hp(y)Γp(y)

Γp((x + y) + 1)

=
Γp (x)hp(y)Γp(y)

Γp(x + y)hp(x + y)

=
hp(y)

hp(x + y)

Γp (x)Γp(y)

Γp(x + y)

=
hp(y)

hp(x + y)
Bp(x , y).

Corollary 3. The relation

Bp(x + 1, y) + Bp(x , y + 1) =
hp(x) + hp(y)

hp(x + y)
Bp(x , y)

holds for all x , y ∈ Zp.

Proof. According to Theorem 3 and Theorem 4 we have

Bp(x + 1, y) + Bp(x , y + 1) =
hp(x)

hp(x + y)
Bp(x , y) +

hp(y)

hp(x + y)
Bp(x , y)



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 221

=
hp(x) + hp(y)

hp(x + y)
Bp(x , y).

Corollary 4. For all x , y ∈ Zp, the equality

Bp(x , y + 1) =
hp(y)

hp(x)
Bp(x + 1, y)

holds.

Proof. It follows from Theorem 3 that

Bp(x , y) =
hp(x + y)

hp(x)
Bp(x + 1, y). (9)

Using (9) in Theorem 4 we obtain that

Bp(x , y + 1) =
hp(y)

hp(x + y)

hp(x + y)

hp(x)
Bp(x + 1, y)

=
hp(y)

hp(x)
Bp(x + 1, y).

Theorem 5. The equality

Bp(x + 1, y + 1) =
hp(x)hp(y)

hp(x + y + 1)hp(x + y)
Bp(x , y)

holds for all x , y ∈ Zp.

Proof. In similar way, we obtain that

Bp(x + 1, y + 1) =
Γp (x + 1)Γp(y + 1)

Γp(x + 1+ y + 1)

=
Γp (x)hp(x)Γp(y)hp(y)

Γp((x + y + 1) + 1)

=
Γp (x)hp(x)Γp(y)hp(y)

Γp(x + y + 1)hp(x + y + 1)

=
hp(x)hp(y)

hp(x + y + 1)

Γp (x)Γp(y)

Γp((x + y) + 1)



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 222

=
hp(x)hp(y)

hp(x + y + 1)

Γp (x)Γp(y)

Γp(x + y)hp(x + y)

=
hp(x)hp(y)

hp(x + y + 1)hp(x + y)
Bp(x , y).

Corollary 5. For all x , y, z ∈ Zp

Bp(x , y)Bp(x + y, z)Bp(x + y + z, w) =
Γp (x)Γp

�

y
�

Γp (z)Γp (w)

Γp

�

x + y + z +w
�

Proof. It is clear from Definition 2 that

Bp(x , y)Bp(x + y, z)Bp(x + y + z, w) =
Γp (x)Γp

�

y
�

Γp

�

x + y
�

Γp

�

x + y
�

Γp (z)

Γp

�

x + y + z
�

Γp

�

x + y + z
�

Γp (w)

Γp

�

x + y + z +w
�

=
Γp (x)Γp

�

y
�

Γp (z)Γp (w)

Γp

�

x + y + z +w
� .

Theorem 6. The equality

Bp(x , 1− x) =

¨

(−1)ℓ(x)+1 if p 6= 2

(−1)σ1(y)+2 if p = 2

holds for all x , y ∈ Zp.

Proof. Note that Γp(1) = −1. By Definition 2 we get

Bp(x , 1− x) =
Γp (x)Γp(1− x)

Γp(x + 1− x)

=
Γp (x)Γp(1− x)

Γp(1)
.

By Proposition 3, if p 6= 2 then

Bp(x , 1− x) = −(−1)ℓ(x) = (−1)ℓ(x)+1

and, if p = 2 then,

Bp(x , 1− x) = −(−1)σ1(y)+1 = (−1)
σ1(y)+2

.

It is well known that the classical beta function can be defined as binomial coefficient
indices. We can give a similar formula for the p-adic beta function.



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 223

Theorem 7. The equality

�

n

k

�

p

Bp(n− k+ 1, k+ 1) =
−1

hp(n+ 1)

holds for all n, k ∈ N, k ≤ n. Here, the notation
�

n
k

�

p
is defined by

�

n

k

�

p

=
(n!)p

((n− k)!)p(k!)p
.

Proof. It is well known that
�

n

k

�

=
n!

(n− k)!k!

for n, k ∈ N, k ≤ n and
(n!)p = (−1)n+1Γp(n+ 1).

Hence, we can write

�

n

k

�

p

Bp(n− k+ 1, k+ 1) =
(n!)p

(k!)p((n− k)!)p

Γp (n− k+ 1)Γp(k+ 1)

Γp(n+ 2)

=
(−1)n+1Γp (n+ 1)

(−1)k+1Γp(k+ 1)(−1)n−k+1Γp(n− k+ 1)

Γp (n− k+ 1)Γp(k+ 1)

Γp(n+ 2)

=
−Γp (n+ 1)

Γp(n+ 2)
.

Thus, by Proposition 1, we obtain that

�

n

k

�

p

Bp(n− k+ 1, k+ 1) =
−Γp (n+ 1)

Γp(n+ 1)hp(n+ 1)

=
−1

hp(n+ 1)
.

Now, we analyze the relationship between the classical beta and the p-adic beta function
at the values of natural numbers.

Theorem 8. The equality

B(n+ 1, m+ 1) = −Bp(n, m)
hp(n)hp(m)

hp(m+ n)(m+ n+ 1)

�

n
p

�

!
�

m
p

�

!
�

m+n
p

�

!
p

�

n
p

�

+
�

m
p

�

−
�

m+n
p

�

holds for all m, n ∈ N.



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 224

Proof. It follows from the definition of classical beta function and main proposition of
classical gamma function that

B(n+ 1, m+ 1) =
Γ(n+ 1)Γ(m+ 1)

Γ(n+m+ 2)

=
Γ(n+ 1)Γ(m+ 1)

(n+m+ 1)Γ(m+ n+ 1)

=
n!m!

(n+m+ 1)(m+ n)!
.

By Proposition 4(i) we have

B(n+ 1, m+ 1) =
(−1)n+1Γp(n+ 1)

�

n
p

�

!p
�

n
p

�

(−1)m+1Γp(m+ 1)
�

m
p

�

!p
�

m
p

�

(n+m+ 1)(−1)m+n+1Γp(m+ n+ 1)
�

m+n
p

�

!p
�

m+n
p

� .

Then, by Proposition 1 we obtain

B(n+ 1, m+ 1) = (−1)
Γp(n)Γp(m)hp(n)hp(m)

Γp(n+m)hp(n+m)

�

n
p

�

!
�

m
p

�

!
�

m+n
p

�

!

p

�

n
p

�

+
�

m
p

�

−
�

m+n
p

�

(n+m+ 1)
.

Thus, using Definition 2 we complete the proof of the theorem

B(n+ 1, m+ 1) = −Bp(n, m)

�

n
p

�

!
�

m
p

�

!
�

m+n
p

�

!

p

�

n
p

�

+
�

m
p

�

−
�

m+n
p

�

(n+m+ 1)

hp(n)hp(m)

hp(n+m)
.

Theorem 9. The equality

B(n+ 1, m+ 1) = Bp(n+ 1, m+ 1)

�

n
p

�

!
�

m
p

�

!
�

m+n+1
p

�

!
p

�

n
p

�

+
�

m
p

�

−
�

m+n+1
p

�

holds for all m, n ∈ N.

Proof. In similar way, using the definitions and Proposition 4(i) we can obtain that

B(n+ 1, m+ 1) =
Γ(n+ 1)Γ(m+ 1)

Γ(n+m+ 2)

=
n!m!

(n+m+ 1)!

=
(−1)n+1Γp(n+ 1)

�

n
p

�

!p
�

n
p

�

(−1)m+1Γp(m+ 1)
�

m
p

�

!p
�

m
p

�

(−1)m+n+2Γp(m+ n+ 2)
�

m+n+1
p

�

!p
�

m+n+1
p

�



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 225

=
Γp(n+ 1)Γp(m+ 1)

Γp(m+ n+ 2)

�

n
p

�

!
�

m
p

�

!
�

m+n+1
p

�

!
p

�

n
p

�

+
�

m
p

�

−[m+n+1
p ]

=Bp(n+ 1, m+ 1)

�

n
p

�

!
�

m
p

�

!
�

m+n+1
p

�

!
p

�

n
p

�

+
�

m
p

�

−[m+n+1
p ]

.

Theorem 10. The equality

B(pn + 1, pm + 1) = Bp(p
n, pm)

(pn−1)!(pm−1)!

(pn−1 + pm−1)!

1

hp(p
n + pm)(pn + pm + 1)

holds for all m, n ∈ N.

Proof. From the definition of classical beta function and main proposition of classical
gamma function follow that

B(pn + 1, pm + 1) =
Γ(pn + 1)Γ(pm + 1)

Γ(pn + pm + 2)
=

pn!pm!

(pn + pm + 1)(pn + pm)!

By Proposition 4 (i) and (ii) we get

B(pn + 1, pm + 1) =
Γp(p

n)(−1)p(pn−1)!ppn−1
Γp(p

m)(−1)p(pm−1)!ppm−1

(pn + pm + 1)Γp(p
n + pm + 1)(−1)pn+pm

�

pn+pm

p

�

!p
�

pn+pm

p

�

Hence, we obtain

B(pn + 1, pm + 1) =
Γp(p

n)Γp(p
m)

Γp(p
n + pm)

(pn−1)!(pm−1)!ppn−1
ppm−1

hp(p
n + pm)(pn−1 + pm−1)!ppn−1+pm−1

(pn + pm + 1)

=Bp(p
n, pm)

(pn−1)!(pm−1)!

(pn−1 + pm−1)!

1

hp(p
n + pm)(pn + pm + 1)

.

Corollary 6. If p 6= 2 then

Bp(
1

2
,
1

2
) =

¨

−1 if p ≡ 3 (mod 4)

1 if p ≡ 1 (mod 4)

Proof. Using Corollary 2 and Proposition 1, we have

Bp(
1

2
,
1

2
) =
Γp(

1
2)Γp(

1
2)

Γp(1)



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 226

=(−1)ℓ(
1
2 )+1, ℓ(

1

2
) = ℓ(

1

2
(p+ 1)) =

1

2
(p+ 1)

=(−1).

¨

1 if p ≡ 3 (mod 4)

−1 if p ≡ 1 (mod 4)

=

¨

−1 if p ≡ 3 (mod 4)

1 if p ≡ 1 (mod 4)
.

Now we prove that the p-adic beta function has the following properties for negative inte-
gers.

Theorem 11. If n, m ∈ N, then

Bp(−n,−m) = (−1)
�

1+
�

n+m
p

�

−
�

n
p

�

−
�

m
p

�� hp(n+m)

hp(n)hp(m)

1

Bp(n, m)

Proof. By Definition 2 and Proposition 2 we get

Bp(−n,−m) =
Γp(−n)Γp(−m)

Γp(−n−m)

=
(−1)n+1−
�

n
p

�

(Γp(n+ 1))−1(−1)m+1−
�

m
p

�

(Γp(m+ 1))−1

(−1)n+m+1−
�

n+m
p

�

(Γp(n+m+ 1))−1
,

and by Proposition 1(i) we have

Bp(−n,−m) =(−1)1+
�

n+m
p

�

−
�

n
p

�

−
�

m
p

� Γp(n+m+ 1)

Γp(n+ 1)Γp(m+ 1)

=(−1)1+
�

n+m
p

�

−
�

n
p

�

−
�

m
p

� Γp(n+m)hp(n+m)

Γp(n)hp(n)Γp(m)hp(m)

=(−1)1+
�

n+m
p

�

−
�

n
p

�

−
�

m
p

� hp(n+m)

hp(n)hp(m)

1

Bp(n, m)
.

Theorem 12. If n, m ∈ N, then

Bp(−n, m) =







(−1)m−
�

n
p

�

+
�−m+n

p

�

hp(n−m)

hp(n)
Bp(n−m, m) if m< n

(−1)n+1−
�

n
p

�

1
hp(n)
(Bp(m− n, n))−1 if n≤ m



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 227

Proof. We know that

Bp(−n, m) =
Γp(−n)Γp(m)

Γp(−n+m)
.

Assume that m< n. Then, by Proposition 2 we can write

Bp(−n, m) =
(−1)n+1−
�

n
p

�

(Γp(n+ 1))−1Γp(m)

(−1)−m+n+1−
�−m+n

p

�

(Γp(−m+ n+ 1))−1

=(−1)n+1−
�

n
p

�

+m−n−1+
�−m+n

p

�Γp(n−m+ 1)Γp(m)

Γp(n+ 1)
.

Using Proposition 1 we obtain

Bp(−n, m) =(−1)m−
�

n
p

�

+
�−m+n

p

�Γp(n−m)hp(n−m)Γp(m)

Γp(n)hp(n)

=(−1)m−
�

n
p

�

+
�−m+n

p

�hp(n−m)

hp(n)
Bp(n−m, m).

Assume that n≤ m. By Proposition 2 we get

Bp(−n, m) =
(−1)n+1−
�

n
p

�

(Γp(n+ 1))−1Γp(m)

Γp(m− n)

=(−1)n+1−
�

n
p

� Γp(m)

Γp(m− n)Γp(n+ 1)
,

and by Proposition 1 we have

Bp(−n, m) =(−1)n+1−
�

n
p

� Γp(m)

Γp(m− n)Γp(n)hp(n)

=
(−1)n+1−
�

n
p

�

hp(n)
(Bp(m− n, n))−1

Theorem 13. If n, m ∈ N then

Bp(n,−m) =









(−1)m+1−[m
p ]

hp(m)
Bp(n−m, m)−1 if m≤ n

(−1)n−[
m
p ]+[m−n

p ]hp(m−n)

hp(m)
Bp(m− n, n) if n< m



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 228

Proof. From Definition 2 we write

Bp(n,−m) =
Γp(n)Γp(−m)

Γp(n−m)
.

If m≤ n, using Proposition 2 we get

Bp(n,−m) =
Γp(n)(−1)m+1−

�

m
p

�

Γp(m+ 1)−1

Γp(n−m)

=(−1)m+1−
�

m
p

� Γp(n)

Γp(m+ 1)Γp(n−m)
.

According to Proposition 1 and Definition 2 we have

Bp(n,−m) =(−1)m+1−
�

m
p

� Γp(n)

Γp(m)hp(m)Γp(n−m)

Bp(n,−m) =
(−1)m+1−
�

m
p

�

hp(m)
Bp(n−m, m)−1.

If n< m, then by Proposition 2 we have

Bp(n,−m) =
Γp(n)(−1)m+1−

�

m
p

�

Γp(m+ 1)−1

(−1)(m−n)+1−
�

m−n
p

�

Γp(m− n+ 1)−1

=(−1)m+1−
�

m
p

�

−(m−n)−1+
�

m−n
p

�Γp(n)Γp(m− n+ 1)

Γp(m+ 1)
.

Using Proposition 1 and Definition 2 we obtain

Bp(n,−m) =(−1)n−
�

m
p

�

+
�

m−n
p

�Γp(n)Γp(m− n)hp(m− n)

Γp(m)hp(m)

Bp(n,−m) =
(−1)n−
�

m
p

�

+
�

m−n
p

�

hp(m− n)

hp(m)
Bp(m− n, n).

3. Conclusions

In the present work we prove that the p-adic beta function Bp : Zp × Zp → Qp has the
following properties:

• If x , y ∈ Zp, then
Bp(x , y) = Bp(y, x)



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 229

• If x , y ∈ Zp, then

Bp(x , y)Bp(x + y, 1− y) =







(−1)ℓ(y)

hp(x)
, p 6= 2

(−1)σ(y)+1

hp(x)
p = 2

• If x , y ∈ Zp, then

Bp(x + 1, y) =
hp(x)

hp(x + y)
Bp(x , y)

• If x , y ∈ Zp, then

Bp(x , y + 1) =
hp(y)

hp(x + y)
Bp(x , y)

• If x , y ∈ Zp, then

Bp(x + 1, y) + Bp(x , y + 1) =
hp(x) + hp(y)

hp(x + y)
Bp(x , y)

• If x , y ∈ Zp, then

Bp(x , y + 1) =
hp(y)

hp(x)
Bp(x + 1, y)

• If x , y ∈ Zp, then

Bp(x + 1, y + 1) =
hp(x)hp(y)

hp(x + y + 1)hp(x + y)
Bp(x , y)

• If x , y, z, w ∈ Zp, then

Bp(x , y)Bp(x + y, z)Bp(x + y + z, w) =
Γp (x)Γp

�

y
�

Γp (z)Γp (w)

Γp

�

x + y + z +w
�

• If x ∈ Zp, then

Bp(x , 1− x) =

¨

(−1)ℓ(x)+1 if p 6= 2

(−1)σ1(y)+2 if p = 2

• If n, k ∈ N, k ≤ n, then
�

n

k

�

p

Bp(n− k+ 1, k+ 1) =
−1

hp(n+ 1)



H. Menken, Ö. Çolakoğlu / Eur. J. Pure Appl. Math, 8 (2015), 214-231 230

• If m, n ∈ N, then

B(n+ 1, m+ 1) = −Bp(n, m)
hp(n)hp(m)

hp(m+ n)(m+ n+ 1)

�

n
p

�

!
�

m
p

�

!
�

m+n
p

�

!
p

�

n
p

�

+
�

m
p

�

−
�

m+n
p

�

• If m, n ∈ N, then

B(n+ 1, m+ 1) = Bp(n+ 1, m+ 1)

�

n
p

�

!
�

m
p

�

!
�

m+n+1
p

�

!
p

�

n
p

�

+
�

m
p

�

−
�

m+n+1
p

�

• If m, n ∈ N, then

B(pn + 1, pm + 1) = Bp(p
n, pm)

(pn−1)!(pm−1)!

(pn−1 + pm−1)!

1

hp(p
n + pm)(pn + pm + 1)

• If p 6= 2, then

Bp(
1

2
,
1

2
) =

¨

−1 if p ≡ 3 (mod 4)

1 if p ≡ 1 (mod 4)

• If m, n ∈ N, then

Bp(−n,−m) = (−1)
�

1+
�

n+m
p

�

−
�

n
p

�

−
�

m
p

�� hp(n+m)

hp(n)hp(m)

1

Bp(n, m)

• If m, n ∈ N, then

Bp(−n, m) =







(−1)m−
�

n
p

�

+
�−m+n

p

�

hp(n−m)

hp(n)
Bp(n−m, m) if m< n

(−1)n+1−
�

n
p

�

1
hp(n)
(Bp(m− n, n))−1 if n≤ m

• If m, n ∈ N, then

Bp(n,−m) =









(−1)m+1−[m
p ]

hp(m)
Bp(n−m, m)−1 if m≤ n

(−1)n−[
m
p ]+[m−n

p ]hp(m−n)

hp(m)
Bp(m− n, n) if n< m

ACKNOWLEDGEMENTS This work is supported by Mersin University. The authors would
like to thank the reviewers for their useful suggestions.



REFERENCES 231

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