Some Properties of the p-Adic Beta Function

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 Çolako˘

glu

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

Abstract. In the present work we consider ap-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 thep-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)∈Zanda bis not divided byp. Thep-adic norm|·|pis defined by

|x|p= ¨

p−vp(x)_{, x}6₌₀

0, x=0.

ByQ_pwe denote the completion of rational numbers fieldQwith respect to the p-adic norm

|·|p. The ring ofp-adic integers is the valuation ring

Z_p=x ∈Q_p:_|x|p≤1 . Note that everyx ∈Z_p can be written in the form

x =b0+b1p+b2p2+. . .+bnpn+. . . ∗_{Corresponding author.}

Email addresses:[email protected](H. Menken),[email protected](Ö. Çolako˘glu)

with 0≤ b_i≤p−1; and also, everyx ∈Q_p can be written in the form

x =b_−n 0p

−n0_{+. . .+b}

0+b1p+b2p2+. . .+bnpn+. . .= X

n≥−n0 b_npn

with 0_≤ b_i≤p₋1 and₋n₀=v_p(x)(for details see[12]).

The classical gamma function is an extension of the factorial function and is defined by the formula

Γ(x) =

∞ Z

0

tx−1e−td 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 integern

(ii) Γ(z+1) =zΓ(z) (Re(z)>0)

(iii) Γ(1−z)Γ(z) = _sinπ_(πz)(Re(z)>0)

(iv) Γ(1₂) =pπ.

It is well known that the classical beta functionB(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 Z

0

tx−1(1−t)y−1d 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)_x+y_y

(iv) B(x,y)B(x+ y, 1₋y) = π

xsin(πy)

(v)

n k

= _(n+1)B(n−1_k+1,k+1) (n,k∈N,k≤n)

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

(viii) B(x,y+1) = y_xB(x+1,y) = _x+y_yB(x,y)

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

where Re(x), Re y, Re(z), Re(w)>0. Thep-adic analogue of the classical gamma function depends on thep-adic version of factorial function. Thep-adic version of factorial function is defined by

(n!)_p:= Y

1≤ j≤n (j,p) =1

j

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

Definition 1([10]). The p-adic gamma functionΓ_p is the continuous extension toZ_pof n7→(−1)n Y

1≤ j<n (j,p) =1

j(n≥2).

Moreover,Γ_p:Z_p→Q_pfunction is defined by Γ_p(x):= lim

n→x(−1)

n Y

1≤ j<n (j,p) =1

j.

According the definition of p-adic factorial function we conclude that: Corollary 1. Γ_p(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 p6=2. ThenΓ_phas the following properties:

(i) For all x∈Z_p

Γ_p(x+1) =h_p(x)Γ_p(x) (1)

where

hp(x):= ¨

−x if |x|p=1

−1 if |x|_p<1

(ii) Γ_p(0) =1,Γ_p(1) =−1,Γ_p(2) =1. For all x∈Z_pwe haveΓ_p(x)

p=1

(iii) For all x,y∈Z_p

Γ_p(x)−Γ_p(y)

p≤ x− y

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

26= 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 forΓ_p(−n) (n∈N)is given by

Γ_p(−n) = (−1)n+1−

”_n

p

—

(Γ_p(n+1))−1. (3) Proposition 3([7, 12]). If p6=2then

Γ_p(x)Γ_p(1−x) = (₋1)ℓ(x)(x∈Z_p) (4)

and for p=2

Γ_p(x)Γ_p(1−x) = (₋1)σ1(x)+1_(x∈_Z

2) (5)

whereℓ:Z_p→1, 2, . . . ,p assigns tox ∈Z_p its residue∈1, 2, . . . ,p modulo pZ_p and whereσ₁is defined by the formula

σ₁(

∞ X

j=0

aj2j) =a1

Corollary 2([12]). Let p6=2. We get Γ_p(1

2)

2_{= (−1)}ℓ(12)

(6)

Nowℓ(1₂) =ℓ(1₂(p+1)) = 1₂(p+1)so that

Γ_p(1

2) 2₌

¨

1 if p≡3 (mod 4)

−1 if p≡1 (mod 4) (7)

Proposition 4([12]). Let n∈Nand let s_n be sum of the digits of n=

s P

j=0

a_jpj (a_s6=0)in base p. Then

(i) Γ_p(n+1) = (₋1)n+1 n!

”_n

p

—

!p[np]

(ii) Γ_p(pn) = (−1)p pn!

pn−1_!ppn−1

(iii) n!= (₋1)n+1−s(₋p)(n−sn)/(p−1) n

Π

j=0Γp €”_n

pj —

(iv) pn!= (₋1)p(₋p)(pn−1)/(p−1) Πn

j=0Γp 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’sp-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 thep-adic beta functions as thep-adic etale andp-adic crys-talline beta functions. Also, some comparisons between thep-adic etale andp-adic crystalline beta functions with relations via Fontaine’s periods are given.

In the present work we study ap-adic analogue of classical beta function by using Morita’s

p-adic gamma function, and we obtain some elemantary properties of thep-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 B_p:Z_p×Z_p→Q_p is defined by the formula

Bp(x,y):=

Γ_p(x) Γ_p(y)

Γ_p(x+y) ,x,y ∈Zp. (8)

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

Theorem 1. The p-adic beta function is symmetric. Namely, B_p(x,y) =B_p(y,x) for x,y∈Z_p.

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

B_p(x,y) =Γp(x) Γp(y) Γ_p(x+y)

=Γp y

Γ_p(x)

Γ_p(y+x)

=B_p(y,x)

Theorem 2. For x,y ∈Z_p, then

B_p(x,y)B_p(x+ y, 1−y) =

 

 (−1)ℓ(y)

hp(x) , p6=2 (₋1)σ₁(y)+1

hp(x) p=2

where

h_p(x):=

¨

−x if |x|_p=1

−1 if |x|p<1

andℓ:Z_p →1, 2, . . . ,p assigns to x ∈Z_p its residue∈1, 2, . . . ,p modulo pZ_p andσ₁ is defined by the formula

σ₁(

∞ X

j=0

a_j2j) =a₁

Proof. Let p6=2. From Definition 2 and Proposition 1 it follows that

B_p(x,y)B_p(x+ y, 1−y) =Γp(x) Γp(y) Γ_p(x+y)

Γ_p x+ yΓ_p(1−y) Γ_p(x+1) =Γp(x) Γp(y)Γp(1− y)

Γ_p(x+1) =Γp(x) Γp(y)Γp(1− y)

Γ_p(x)h_p(x)

=Γp(y)Γp(1−y) h_p(x) ,

and by Proposition 3 we obtain that

B_p(x,y)B_p(x+y, 1− y) =(−1) ℓ(y)

h_p(x) .

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

Theorem 3. The equality

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

h_p(x+y)Bp(x,y) holds for all x,y ∈Z_p.

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

=Γp(x)hp(x)Γp(y) Γ_p((x+y) +1) = Γp(x)hp(x)Γp(y)

Γ_p(x+ y)h_p(x+ y)

= hp(x) hp(x+y)

Γ_p(x) Γ_p(y)

Γ_p(x+y)

= hp(x)

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

Theorem 4. The equality

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

B_p(x,y)

holds for all x,y ∈Z_p.

Proof. From Definition 2 and Proposition 1 we get

B_p(x,y+1) =Γp(x) Γp(y+1) Γ_p(x+ y+1) =Γp(x)hp(y)Γp(y)

Γ_p((x+y) +1) = Γp(x)hp(y)Γp(y)

Γ_p(x+ y)h_p(x+ y)

= hp(y) hp(x+y)

Γ_p(x) Γ_p(y)

Γ_p(x+y)

= hp(y)

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

Corollary 3. The relation

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

h_p(x+y) Bp(x,y) holds for all x,y ∈Z_p.

Proof. According to Theorem 3 and Theorem 4 we have

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

h_p(x+y)Bp(x,y) +

h_p(y)

= hp(x) +hp(y) hp(x+y)

B_p(x,y).

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

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

B_p(x+1,y)

holds.

Proof. It follows from Theorem 3 that

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

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

Using (9) in Theorem 4 we obtain that

Bp(x,y+1) =

h_p(y)

h_p(x+y)

h_p(x+y)

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

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

Theorem 5. The equality

Bp(x+1,y+1) =

h_p(x)h_p(y)

h_p(x+ y+1)h_p(x+ y)Bp(x,y) holds for all x,y ∈Z_p.

Proof. In similar way, we obtain that

B_p(x+1,y+1) =Γp(x+1) Γp(y+1) Γ_p(x+1+ y+1) =Γp(x)hp(x)Γp(y)hp(y)

Γ_p((x+y+1) +1) = Γp(x)hp(x)Γp(y)hp(y)

Γ_p(x+y+1)h_p(x+y+1) = hp(x)hp(y)

hp(x+y+1)

Γ_p(x) Γ_p(y)

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

Γ_p(x) Γ_p(y)

Γ_p(x+ y)h_p(x+ y)

= hp(x)hp(y)

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

Corollary 5. For all x,y,z∈Z_p

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

Γ_p(z) Γ_p(w)

Γ_p x+ y+z+w

Proof. It is clear from Definition 2 that

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

Γ_p x+y

Γ_p x+ yΓ_p(z)

Γ_p x+ y+z

Γ_p x+ y+zΓ_p(w)

Γ_p x+ y+z+w

=Γp(x) Γp y

Γ_p(z) Γ_p(w)

Γ_p x+y+z+w .

Theorem 6. The equality

B_p(x, 1−x) =

¨

(−1)ℓ(x)+1 if p6=2

(−1)σ1(y)+2 _{if p=2} holds for all x,y ∈Z_p.

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

B_p(x, 1−x) =Γp(x) Γp(1−x) Γ_p(x+1−x) =Γp(x) Γp(1−x)

Γ_p(1) .

By Proposition 3, ifp6=2 then

B_p(x, 1−x) =−(−1)ℓ(x)= (−1)ℓ(x)+1

and, ifp=2 then,

Bp(x, 1−x) =−(−1)σ1(y)+1= (−1)

σ1(y)+2

.

Theorem 7. The equality

_n

k

‹

p

B_p(n−k+1,k+1) = −1 hp(n+1)

holds for all n,k∈N,k≤n. Here, the notation n_kp 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! forn,k∈N,k≤nand

(n!)_p= (₋1)n+1Γ_p(n+1). Hence, we can write

_n

k

‹

p

B_p(n−k+1,k+1) = (n!)p (k!)_p((n₋k)!)_p

Γ_p(n−k+1) Γ_p(k+1) Γ_p(n+2)

= (−1)

n+1_Γ

p(n+1)

(₋1)k+1_Γ

p(k+1)(−1)n−k+1Γp(n−k+1)

Γ_p(n−k+1) Γ_p(k+1) Γ_p(n+2) =−Γp(n+1)

Γ_p(n+2) .

Thus, by Proposition 1, we obtain that _n

k

‹

p

Bp(n−k+1,k+1) =

−Γ_p(n+1) Γ_p(n+1)h_p(n+1) = −1

h_p(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)

h_p(m+n)(m+n+1)

”_n p —

!”m_p—! ”_m+n

p —

!

p

”_n

p

—

+”m_p—−”m+n p

—

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+1_Γ

p(n+1) ”_n p — !p ”_n p —

(₋1)m+1_Γ

p(m+1) ”_m p — !p ”_m p —

(n+m+1)(−1)m+n+1_Γ

p(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) Γ_p(n+m)h_p(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) =₋B_p(n,m)

”_n p —

!”m_p—! ”_m+n p — ! p ”_n p —

+”m_p—−”m+n p

—

(n+m+1)

h_p(n)h_p(m)

hp(n+m) .

Theorem 9. The equality

B(n+1,m+1) =B_p(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+1_Γ

p(n+1) ”_n p — !p ”_n p —

(₋1)m+1_Γ

p(m+1) ”_m p — !p ”_m p —

(−1)m+n+2_Γ

p(m+n+2)

”_m+n+1 p

— !p

”_m+n+1

p

=Γp(n+1)Γp(m+1) Γ_p(m+n+2)

”_n p —

!”m_p—! ”_m+n+1

p —

!

p

”_n

p

—

+”m_p—−[m+pn+1]

=B_p(n+1,m+1)

”_n p —

!”m_p—! ”_m+n+1

p —

!

p

”_n

p

—

+”m_p—−[m+pn+1] .

Theorem 10. The equality

B(pn+1,pm+1) =B_p(pn,pm)(p

n−1_)!(pm−1_)!

(pn−1_+pm−1_)!

1

hp(pn+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(p

n_+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_Γ

p(pm)(−1)p(pm−1)!pp m−1

(pn_+pm_+1)Γ

p(pn+pm+1)(−1)p

n_+pm”pn+pm p

— !p

”pn+pm p

—

Hence, we obtain

B(pn+1,pm+1) =Γp(p

n_)Γ p(pm)

Γ_p(pn_+pm₎

(pn−1)!(pm−1)!ppn−1ppm−1 h_p(pn_+pm_)(pn−1_+pm−1_)!ppn−1_+pm−1_(pn+

pm₊₁₎

=B_p(pn,pm)(p

n−1_)!(pm−1_)!

(pn−1_+pm−1_)!

1

h_p(pn_+pm_)(pn_+pm₊₁₎.

Corollary 6. If p6=2then

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

B_p(1

2, 1 2) =

Γ_p(1₂)Γ_p(1₂)

=(−1)ℓ(12)+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 thep-adic beta function has the following properties for negative inte-gers.

Theorem 11. If n,m∈N, then

B_p(−n,−m) = (−1)

”

1+”n+_pm—−”n p

—

−”m p

—— _h

p(n+m)

h_p(n)h_p(m)

1

B_p(n,m) Proof. By Definition 2 and Proposition 2 we get

B_p(−n,−m) =Γp(−n)Γp(−m) Γ_p(₋n−m)

=(−1)

n+1−”n p

—

(Γ_p(n+1))−1(−1)m+1−

”_m

p

—

(Γ_p(m+1))−1 (₋1)n+m+1−

”_n+m

p

—

(Γ_p(n+m+1))−1

,

and by Proposition 1(i) we have

Bp(−n,−m) =(−1)

1+”n+_pm—−”n p

—

−”m p

— Γ

p(n+m+1)

Γ_p(n+1)Γ_p(m+1) =(−1)1+

”_n+m

p

—

−”n p

—

−”m p

— Γ

p(n+m)hp(n+m)

Γ_p(n)h_p(n)Γ_p(m)h_p(m)

=(₋1)1+

”_n+m

p

—

−”n p

—

−”m p

— h

p(n+m)

h_p(n)h_p(m)

1

B_p(n,m).

Theorem 12. If n,m∈N, then

B_p(−n,m) =

 



(₋1)m−

”_n

p

—

+”−m_p+n—hp(n−m)

hp(n) Bp(n−m,m) if m<n

(₋1)n+1−

”_n

p

—

1

hp(n)(Bp(m−n,n))

Proof. We know that

B_p(−n,m) = Γp(−n)Γp(m) Γ_p(₋n+m) .

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

Bp(−n,m) =

(₋1)n+1−

”_n

p

—

(Γ_p(n+1))−1_Γ

p(m)

(₋1)−m+n+1−

”_−m+n

p

—

(Γ_p(₋m+n+1))−1

=(₋1)n+1−

”_n

p

—

+m−n−1+”−m_p+n—Γp(n−m+1)Γp(m)

Γ_p(n+1) .

Using Proposition 1 we obtain

B_p(−n,m) =(−1)m−

”_n

p

—

+”−m_p+n—Γp(n−m)hp(n−m)Γp(m)

Γ_p(n)h_p(n)

=(₋1)m−

”_n

p

—

+”−m_p+n—hp(n−m)

h_p(n) Bp(n−m,m).

Assume thatn≤m. By Proposition 2 we get

B_p(−n,m) =(−1)

n+1−”n p

—

(Γ_p(n+1))−1Γ_p(m) Γ_p(m₋n)

=(₋1)n+1−

”_n

p

— Γ

p(m)

Γ_p(m−n)Γ_p(n+1),

and by Proposition 1 we have

B_p(−n,m) =(−1)n+1−

”_n

p

— Γ

p(m)

Γ_p(m₋n)Γ_p(n)h_p(n)

=(−1)

n+1−”n p

—

hp(n)

(B_p(m₋n,n))−1

Theorem 13. If n,m∈Nthen

Bp(n,−m) =   

 

(−1)m+1−[mp]

hp(m) Bp(n−m,m)

−1 _{if m≤n}

(−1)n−[mp]+[m−pn]_h p(m−n)

Proof. From Definition 2 we write

B_p(n,−m) = Γp(n)Γp(−m) Γ_p(n₋m) .

Ifm≤n, using Proposition 2 we get

B_p(n,−m) =Γp(n)(−1)

m+1−”m p

—

Γ_p(m+1)−1 Γ_p(n₋m)

=(₋1)m+1−

”_m

p

— Γ

p(n)

Γ_p(m+1)Γ_p(n−m).

According to Proposition 1 and Definition 2 we have

B_p(n,₋m) =(₋1)m+1−

”_m

p

— Γ

p(n)

Γ_p(m)h_p(m)Γ_p(n−m)

B_p(n,−m) =(−1)

m+1−”m p

—

h_p(m) Bp(n−m,m)

−1_.

Ifn<m, then by Proposition 2 we have

B_p(n,−m) = Γp(n)(−1)

m+1−”m p

—

Γ_p(m+1)−1 (−1)(m−n)+1−

”_m−n

p

—

Γ_p(m−n+1)−1

=(−1)m+1−

”_m

p

—

−(m−n)−1+”m−n_p —Γp(n)Γp(m−n+1)

Γ_p(m+1) .

Using Proposition 1 and Definition 2 we obtain

B_p(n,−m) =(−1)n−

”_m

p

—

+”m−n_p —Γp(n)Γp(m−n)hp(m−n)

Γ_p(m)h_p(m)

Bp(n,−m) =

(₋1)n−

”_m

p

—

+”m−n_p —

h_p(m₋n)

h_p(m) Bp(m−n,n).

3. Conclusions

In the present work we prove that the p-adic beta function B_p : Z_p×Z_p → Q_p has the following properties:

• If x,y∈Z_p, then

• If x,y∈Z_p, then

B_p(x,y)B_p(x+ y, 1−y) =

 

 (−1)ℓ(y)

hp(x) , p6=2 (₋1)σ(y)+1

hp(x) p=2

• If x,y∈Z_p, then

Bp(x+1,y) =

h_p(x)

h_p(x+ y)Bp(x,y) • If x,y∈Z_p, then

Bp(x,y+1) =

hp(y)

h_p(x+ y)Bp(x,y) • If x,y∈Z_p, then

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

B_p(x,y)

• If x,y∈Z_p, then

Bp(x,y+1) =

hp(y)

h_p(x)Bp(x+1,y) • If x,y∈Z_p, then

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

B_p(x,y)

• If x,y,z,w∈Z_p, then

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

Γ_p(z) Γ_p(w)

Γ_p x+y+z+w

• If x∈Z_p, then

Bp(x, 1−x) = ¨

(₋1)ℓ(x)+1 _{if p6=2}

(−1)σ1(y)+2 _{if p=2} • Ifn,k∈N,k≤n, then

_n

k

‹

p

• Ifm,n∈N, then

B(n+1,m+1) =₋B_p(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

—

• Ifm,n_∈N, then

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

”_n p —

!”m_p—! ”_m+n+1

p — ! p ”_n p —

+”m_p—−”m+n+1

p

—

• Ifm,n∈N, then

B(pn+1,pm+1) =B_p(pn,pm)(p

n−1_)!(pm−1_)!

(pn−1_+pm−1_)!

1

hp(pn+pm)(pn+pm+1)

• Ifp6=2, then

B_p(1

2, 1 2) =

¨

−1 ifp≡3 (mod 4)

1 ifp≡1 (mod 4) • Ifm,n∈N, then

B_p(−n,−m) = (−1)

”

1+”n+_pm—−”n p

—

−”m p

—— h

p(n+m)

hp(n)hp(m) 1

Bp(n,m)

• Ifm,n∈N, then

Bp(−n,m) =  



(−1)m−

”_n

p

—

+”−m_p+n—hp(n−m)

hp(n) Bp(n−m,m) ifm<n

(−1)n+1−

”_n

p

—

1

hp(n)(Bp(m−n,n))

−1 _ifn≤m

• Ifm,n∈N, then

B_p(n,−m) =

  

 

(−1)m+1−[mp]

hp(m) Bp(n−m,m)

−1 _ifm≤n

(₋1)n−[mp]+[m−pn]_h p(m−n)

hp(m) Bp(m−n,n) ifn<m

References

[1] L. C. Andrews. Special Functions for Engineer and Applied mathematicians, Macmillan Publishing Company, London, 1985.

[2] F. Baldassarri.Etale and crystalline beta and gamma functions via Fontaine’s periods, Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematicae Applicazioni (9), 17(2), 175-198. 2006.

[3] D. Barsky. On Morita’s p-adic gamma function, Mathematical Proceedings of the Cam-bridge Philosophical Society, 89(1), 23-27. 1981.

[4] M. Boyarsky. p-adic gamma functions and Dwork cohomology, Transactions of the Ameri-can Mathematical Society, 257(2), 359-369. 1980.

[5] H. Cohen and E. Friedman.Raabe’s formula for p-adic gamma and zeta functions, Univer-sitÃl’ de Grenoble. Annales de l’Institut Fourier, 58(1), 363-376. 2008.

[6] J. Diamond.The p-adic log gamma function and p-adic Euler constant, Transactions of the American Mathematical Society, 233, 321-337. 1977.

[7] B. Dwork. A note on p-adic gamma function, Study group on ultrametric analysis, 9th year: 1981/82, No. 3 (Marseille, 1982), Exp. No. J5, 10 pp., Inst. Henri Poincaré, Paris, 1983.

[8] B. H. Gross and N. Koblitz.Gauss Sums and the p-adicΓ-function, The Annals of Mathe-matics, Second Series, 109(3), 569-581. 1979.

[9] T. K. Kim. A note on analogue of gamma functions, in Proceedings of the 5th Transcen-dental Number Theory, 111–118, Gakushin University, Tokyo, Japan, 1997.

[10] Y. Morita.A p-adic analogue of theΓ-function, Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 22, 225-266. 1975.

[11] G. Overholtzer.Sum functions in elementary p-adic analysis, American Journal of Mathe-matics, 74, 332-346. 1952.

[12] W. H. Schikhof.Ultrametric Calculus: An Introduction to p-adic Analysis, Cambridge Uni-versity Press, 1984.