generalized p -adic hypergeometric functions

LXVII.2 (1994)

On special values of

generalized p -adic hypergeometric functions

by

Kaori Ota (Tokyo)

We generalize results for ratios of generalized hypergeometric functions obtained by P. T. Young [9] to the case of a power of p. More precisely, we generalize Theorems 3.2 and 3.4 in [9]. In order to do that, introducing a generalization of the p-adic Gamma function to a power of p seems to simplify expressions of the results, and this is done in Section 1. There we prove properties of this generalized Gamma function. In Section 2, after proving some preliminary results for certain ratios of binomial coefficients and giving a “truncated” version of Dwork’s theorem, we prove the theorems.

We follow very closely the proofs of the corresponding theorems in [9], and our methods also apply to the well-poised series (Theorems 3.1 and 3.3 in [9]) treated in [10]. Some applications are also given in [10].

Throughout this paper, p is a prime, ν

_p

denotes the p-adic valuation normalized by ν

p

(p) = 1 and by the p-adic expansion we mean the standard p-adic expansion whose coefficients are all in {0, 1, . . . , p − 1}.

1. The p-adic Gamma function is defined by Γ

_p

(n + 1) = (−1)

ⁿ⁺¹

Y

n j=1 p - j

j

for a positive integer n and is extended by continuity to all of Z

_p

: Γ

p

: Z

p

→ Z

^∗_p

(cf. [6]). Here Z

p

is the ring of p-adic integers.

For a non-negative integer l, we define a map h

_l

from Z

_p

to Z

_p

by h

_l

(x) = X

j≥l

x

_j

p

^j−l

for x ∈ Z

p

with the p-adic expansion P

_∞

j=0

x

j

p

^j

. Then h

l

is obviously con- tinuous. Let t be a positive integer and q = p

^t

.

[141]

Definition. For x ∈ Z

p

, we define a map Γ

q

from Z

p

to Z

^∗_p

by Γ

_q

(x + 1) =

t−1

Y

l=0

Γ

_p

(h

_l

(x) + 1).

Then Γ

_q

is continuous and, for t = 1, Γ

_q

coincides with Γ

_p

. N. Koblitz con- structed and studied an extension of the p-adic Gamma function (cf. [4], [5]).

There he uses Γ

_q

for a different function, but this will cause no confusion.

Γ

_q

has the following properties analogous to Γ

_p

.

Proposition 1. (1) If x ≡ y (mod p

ⁿ

) for n ≥ t − 1, then Γ

_q

(x) ≡ Γ

_q

(y) (mod p

^n−(t−1)

).

(2) For a positive integer n,

Γ

_q

(n + 1) = (−1)

^A

p

^−B

Y

n j=1 q - j

j,

where

A = t + X

t−1

i=0

 n p

ⁱ



and B =

X

t i=1

 n p

ⁱ



− t

 n p

^t

 , and [x] denotes the largest integer not exceeding x.

(3) We have Γ

q

(x + 1) =

 (−1)

^k+1

h

_k

(x)Γ

_q

(x) if x 6∈ qZ

_p

and k = ν

_p

(x), (−1)

^t

Γ

_q

(x) if x ∈ qZ

_p

.

(4) For p odd,

Γ

_q

(x)Γ

_q

(1 − x) = (−1)

^t−1+Rt(x)

,

where R

_t

(x) is the integer in {1, 2, . . . , q} satisfying R

_t

(x) ≡ x (mod q). For p = 2,

Γ

_q

(x)Γ

_q

(1 − x) =

 (−1)

^t+1+xt

if x 6∈ qZ

p

and x 6∈ Z

^∗_p

, (−1)

^t+xt

if x ∈ qZ

_p

or x ∈ Z

^∗_p

, where x

_t

is the coefficient of p

^t

in the p-adic expansion of x.

(5) Let N be an integer ≥ 2, prime to p. Then

N −1

Y

i=0

Γ

_q

 x + i N



= Γ

_q

(x)

N −1

Y

i=1

Γ

_q

 i N



g

_N

(x)

⁻¹

,

where g

_N

(x) = N

^Rt(x)−1

N

^{(q−1)xR0t(x)}

with R

⁰_t

(x) = (x − R

_t

(x))/q.

(6) For Gauss sums, we have the following. For p odd and j in {1, 2, . . . . . . , q − 2},

Γ

q

 j 1 − q + 1



= (−1)

^t

q

⁻¹

π

^s(j)

τ

q

(ω

_q^j

, ψ

π,q

).

Here τ

_q

is the Gauss sum defined by τ

q

(χ, ψ) = − X

x∈F_q

χ(x)ψ(x),

π

^p−1

= −p, ψ

_π,q

(x) = ζ

π^tr(x)

is the additive character of F

_q

with the p-th root of unity ζ

π

satisfying ζ

π

≡ 1 + π (mod π

²

), ω

q

is the Teichm¨uller character , and s(j) = P

_t−1

i=0

j

_i

for the p-adic expansion j = P

_t−1

i=0

j

_i

p

ⁱ

. P r o o f. We use the properties of Γ

_p

(cf. [6], Chap. 14).

(1) If x ≡ y (mod p

ⁿ

), then Γ

_p

(x) ≡ Γ

_p

(y) (mod p

ⁿ

), so this is obvious.

(2) We have Y

n

j=1 q - j

j = Y

n j=1 p - j

j

[n/p]

Y

j=1 p - j

(pj) . . .

[n/p

Y

^t−1] j=1 p - j

(p

^t−1

j)

= (−1)

ⁿ⁺¹

Γ

p

(n + 1)(−1)

^[n/p]+1

p

^{[n/p]−[n/p2]}

Γ

p

([n/p] + 1) . . . . . . (−1)

^[n/pt−1]+1

(p

^t−1

)

^{[n/pt−1]−[n/pt]}

Γ

_p

([n/p

^t−1

] + 1).

Since

t−1

Y

i=0

Γ

p

([n/p

ⁱ

] + 1) =

t−1

Y

i=0

Γ

p

(h

i

(n) + 1) = Γ

q

(n + 1), the result follows.

(3) Γ

_p

satisfies the following functional equation:

Γ

_p

(x + 1) =

 −xΓ

_p

(x) if x ∈ Z

^∗_p

,

−Γ

p

(x) if x ∈ pZ

p

.

Set f (x) to be Γ

q

(x + 1). When k = ν

p

(x + 1) < ∞, let x + 1 = P

j≥k

y

j

p

^j

be the p-adic expansion of x + 1. Then for l ≤ k, we have

h

_l

(x) = (p − 1) + . . . + (p − 1)p

^k−1−l

+ (y

_k

− 1)p

^k−l

+ y

_k+1

p

^k+1−l

+ . . .

= − 1 + h

_l

(x + 1).

For l > k, we have h

_l

(x) = h

_l

(x + 1).

1. If k ≥ t, then

f (x + 1) = (−1)

^t

t−1

Y

l=0

Γ

_p

(h

_l

(x) + 1) = (−1)

^t

f (x).

This also holds for k = ∞.

2. If k < t, then

f (x + 1) = (−1)

^k+1

h

k

(x + 1)

t−1

Y

l=0

Γ

p

(h

l

(x) + 1) = (−1)

^k+1

h

k

(x + 1)f (x).

By substituting x − 1 for x, the result follows.

(4) Γ

p

satisfies the following:

Γ

p

(x)Γ

p

(1 − x) =

 (−1)

^R1(x)

if p > 2, ε(x) if p = 2, where

ε(x) =

 −1 if x ≡ 0, 1 (mod 4), 1 if x ≡ 2, 3 (mod 4).

Assume that k = ν

p

(x) < ∞ and let x = P

_∞

l=k

x

l

p

^l

be the p-adic expansion of x. Then for l ≤ k, we have

h

_l

(−x) = (p − x

_k

)p

^k−l

+ X

j≥k+1

(p − x

_j

− 1)p

^j−l

= −h

_l

(x).

For l > k, we have

h

l

(−x) = −1 − h

l

(x).

1. If k ≥ t, then

Γ

_q

(x + 1)Γ

_q

(1 − x) =

t−1

Y

l=0

{Γ

_p

(h

_l

(x) + 1)Γ

_p

(−h

_l

(x) + 1)}

= (−1)

^t

t−1

Y

l=0

{Γ

_p

(h

_l

(x))Γ

_p

(1 − h

_l

(x))}

=

 (−1)

^t

Q

_t−1

l=0

(−1)

^p

= 1 for p > 2, (−1)

^t

Q

_t−1

l=0

ε(h

_l

(x)) for p = 2.

Since Γ

_q

(x + 1) = (−1)

^t

Γ

_q

(x), we have Γ

_q

(x)Γ

_q

(1 − x) =

 (−1)

^t

if p > 2, (−1)

^t+xt

if p = 2.

This also holds for k = ∞.

2. If k < t, then Γ

_q

(x + 1)Γ

_q

(1 − x)

=

t−1

Y

l=0

Γ

_p

(h

_l

(x) + 1) Y

k l=0

Γ

_p

(−h

_l

(x) + 1)

t−1

Y

l=k+1

Γ

_p

(−h

_l

(x))

= n

^t−1

Y

l=0

Γ

p

(h

l

(x) + 1) o

(−1)

^k+1

(−h

k

(x))

t−1

Y

l=0

Γ

p

(−h

l

(x))

=

 (−1)

^k

h

_k

(x)(−1)

^{k+(p−xk)+...+(p−xt−1)}

if p > 2, (−1)

^k

h

k

(x) Q

_t−1

l=0

ε(−h

l

(x)) if p = 2.

Since Γ

_q

(x + 1) = (−1)

^k+1

h

_k

(x)Γ

_q

(x), we have

Γ

_q

(x)Γ

_q

(1 − x) =

 



(−1)

^t−1+Rt(x)

if p > 2,

(−1)

^t+xt

if p = 2 and x ∈ Z

^∗_p

, (−1)

^t−1+xt

if p = 2 and x 6∈ Z

^∗_p

. (5) Set

g

_N

(x) =

N −1

Y

i=1

Γ

_q

 i N



_{N −1}

Y

i=0

Γ

_q

 x + i N



₋₁

Γ

_q

(x).

Then for non-zero x, g

N

(x + 1)

g

_N

(x) = Γ

q

(x + 1)Γ

q

(x/N ) Γ

_q

(x)Γ

_q

((x + N )/N )

=

 h

_k

(x)/h

_k

(x/N ) if x 6∈ qZ

_p

and ν

_p

(x) = k,

1 if x ∈ qZ

p

.

If ν

_p

(x) = k, then h

_k

(x) = N · h

_k

(x/N ), so g

_N

(x + 1)

g

N

(x) =

 N if x 6∈ qZ

p

, 1 if x ∈ qZ

_p

.

By continuity in x, we only need to consider positive integers. Hence g

_N

(n) =

 N · g

_N

(n − 1) if n − 1 6∈ qZ

_p

, g

N

(n − 1) if n − 1 ∈ qZ

p

= N

n−1−[(n−1)/q]

g

N

(1).

As g

N

(1) = 1 and [(n − 1)/q] = R

⁰_t

(n), we get the result.

(6) By the Gross–Koblitz formula (cf. [6], Chap. 15), (−1)

^t

q

⁻¹

π

^s(j)

τ

_q

(ω

_q^j

, ψ

_π,q

) =

t−1

Y

l=0

Γ

_p

 1 −

 p

^l

j q − 1



. But

1 −

 p

^t−l

j q − 1



= 1 + h

l

 j 1 − q



,

so the result follows.

R e m a r k s. 1. Just like Γ

p

,

N −1

Y

i=1

Γ

q

 i N



=

n ±1 if N is odd,

±1 or ± i if N is even.

2. For 0 < a, b < a + b < q − 1,

J(ω

_q^−a

, ω

_q^−b

) = τ

q

(ω

^−a_q

, ψ

π,q

)τ

q

(ω

^−b_q

, ψ

π,q

) τ

q

(ω

^−(a+b)q

, ψ

π,q

)

= (−p)

^νp

((

^a+bb

)) Γ

^{q q−1a}

Γ

_{q q−1}^b

Γ

_q ^a+b_q−1

, where J(χ

1

, χ

2

) denotes the Jacobi sum. This follows from (6).

2. We generalize theorems in [9] to a power of p. Let p be an odd prime and let q denote p

^t

as before.

Proposition 2. Let a and b be integers satisfying 0 < a < b < q − 1.

For r ≥ 0, set

n

_r

= b q

^r

− 1

q − 1 and m

_r

= a q

^r

− 1 q − 1 . Then for r > 0,

 n

_r

m

r



 n

_r−1

m

r−1

 = (−1)

^t

(−p)

^νp

((

a^b

)) Γ

_q

(1 + n

_r

)

Γ

q

(1 + m

r

)Γ

q

(1 + n

r

− m

r

)

= (−p)

^νp

((

^b_a

)) Γ

^q

(−m

r

)Γ

q

(m

r

− n

r

) Γ

_q

(−n

_r

) . P r o o f. From Proposition 1(2), we have

Γ

q

(1 + n

r

) = (−1)

^{t+Σt−1i=0[nr/pi]}

p

^{−(Σti=1[nr/pi]−t[nr/pt])}

n_r

Y

j=1 q - j

j,

and from

n_r

Y

j=1

j =

 Y

^nr

j=1 q - j

j



ⁿ

Y

^r−1

j=1

(qj)

 ,

we have

n

_r

!

n

r−1

! = q

^nr−1

nr

Y

j=1 q - j

j.

Hence  n

r

m

_r



 n

r−1

m

_r−1

 =  q

^nr−1

n_r

Y

j=1 q - j

j



q

^mr−1

m_r

Y

j=1 q - j

j



₋₁



q

^{nr−1−mr−1}

n_r

Y

−m_r j=1 q - j

j



₋₁

= (−1)

^e1

p

^e2

Γ

_q

(1 + n

_r

)

Γ

_q

(1 + m

_r

)Γ

_q

(1 + n

_r

− m

_r

) , where

e

₁

= X

t−1 i=0

 n

_r

p

ⁱ

 +

X

t−1 i=0

 m

_r

p

ⁱ

 +

X

t−1 i=0

 n

_r

− m

_r

p

ⁱ

 + 3t,

e

2

= X

t i=1

 n

r

p

ⁱ



− X

t i=1

 m

r

p

ⁱ



− X

t

i=1

 n

_r

− m

_r

p

ⁱ



− t

 n

_r

p

^t

 + t

 m

_r

p

^t

 + t

 n

_r

− m

_r

p

^t

 . From simple calculations, we can show that e

₁

≡ ν

_p ^b_a

+ t (mod 2), and e

2

= ν

p b

a

.

Also from Proposition 1(4),

Γ

_q

(1 + n

_r

) = (−1)

^{t−1+Rt(−nr)}

Γ

_q

(−n

_r

)

⁻¹

= (−1)

^t−1+q−b

Γ

_q

(−n

_r

)

⁻¹

and similar identities hold, so we get the result.

The following corollary has been given in [10] in a more general form.

Corollary 3 ([10], Theorem 2.2 and Corollary 2.3). With the same notations and assumptions as in Proposition 2, for r > 0,

 n

_r

m

r



 n

_r−1

m

r−1

 ≡ J(ω

^−a_q

, ω

_q^a−b

) (mod p

^{tr−(t−1)+νp}

((

a^b

))).

P r o o f. Since

−n

_r

≡ b

q − 1 (mod q

^r

), −m

_r

≡ a

q − 1 (mod q

^r

) and

m

r

− n

r

≡ b − a

q − 1 (mod q

^r

), from Proposition 1(1) we get

Γ

_q

(−n

_r

) ≡ Γ

_q

 b q − 1



(mod p

^tr−(t−1)

), and so on.

Hence the result follows from Remark 2 after Proposition 1.

R e m a r k. From Proposition 2, for n

r

= b q

^r

− 1

q − 1 and m

r

= a q

^r

− 1 q − 1 with 0 < a < b < q − 1,

ν

_p

 n

_r

m

_r



= r · ν

_p

 b a



. Note also that from ν

p

(n!) = (n − s(n))/(p − 1),

ν

_p

 b a



= 0 ⇔ s(b) = s(a) + s(b − a).

As in [1] and [9], for α in Z

p

we define α

⁰

and µ

α

as follows:

pα

⁰

− α = µ

α

∈ {0, 1, . . . , p − 1}.

Also for i ≥ 1, we define

α

⁽ⁱ⁾

= (α

⁽ⁱ⁻¹⁾

)

⁰

and µ

⁽ⁱ⁾_α

= µ

_α(i)

. Then the following properties hold:

1. If −α = P

_∞

i=0

a

i

p

ⁱ

is the p-adic expansion of −α, then µ

⁽ⁱ⁾_α

= a

_i

and α

⁽ⁱ⁾

= −

X

∞ j=i

a

_j

p

^j−i

= −h

_i

(−α).

2. For α = a/(q − 1) with 0 ≤ a = P

_t−1

i=0

a

_i

p

ⁱ

≤ q − 1, µ

⁽ⁱ⁾_α

= a

_j

, α

⁽ⁱ⁾

= a

_j

+ a

_j+1

p + . . . + a

_j−1

p

^t−1

q − 1 if i ≡ j (mod t)

with j ∈ {0, 1, . . . , t − 1}, and α

^(t)

= α.

3. For n

_r

= a(q

^r

− 1)/(q − 1) with 0 ≤ a ≤ q − 1, (−n

_r

)

^(t)

= −n

_r−1

. The generalized hypergeometric function

k

F

l

(α

1

, α

2

, . . . , α

k

; β

1

, β

2

, . . . . . . , β

_l

; X) is defined by

k

F

_l

 α

₁

, α

₂

, . . . , α

_k

β

₁

, β

₂

, . . . , β

_l

; X



= X

∞ s=0

(α

₁

)

_s

(α

₂

)

_s

. . . (α

_k

)

_s

(β

₁

)

_s

(β

₂

)

_s

. . . (β

_l

)

_s

· s! X

^s

, where (α)

_n

= α(α + 1) . . . (α + n − 1) for n > 0 and (α)

₀

= 1.

For i ≥ 0, define

k

F

_l⁽ⁱ⁾

 α

₁

, α

₂

, . . . , α

_k

β

1

, β

2

, . . . , β

l

; X



=

_k

F

_l

 α

⁽ⁱ⁾₁

, α

⁽ⁱ⁾₂

, . . . , α

⁽ⁱ⁾_k

β

₁⁽ⁱ⁾

, β

₂⁽ⁱ⁾

, . . . , β

_l⁽ⁱ⁾

; X

 . Also as in [9], we denote by

k

F

_l^(t)

 α

₁

, α

₂

, . . . , α

_k

β

1

, β

2

, . . . , β

l

; X



the analytic element which extends the ratio

k

F

l

 α

1

, α

2

, . . . , α

k

β

₁

, β

₂

, . . . , β

_l

; X



k

F

_l^(t)

 α

₁

, α

₂

, . . . , α

_k

β

1

, β

2

, . . . , β

l

; X

^q

 .

For a series F (X) = P

_∞

s=0

A(s)X

^s

, set F

n

(X) = P

_pⁿ₋₁

s=0

A(s)X

^s

.

Proposition 4. Let α and β be elements of Z

_p

such that if −α = P

_∞

i=0

a

_i

p

ⁱ

and −β = P

_∞

i=0

b

_i

p

ⁱ

are p-adic expansions, then (4.1) a

_i

+ b

_i

< p for any i.

Then

2

F

₁^(t)

 α, β 1 ; 1



= Γ

q

(α)Γ

q

(β) Γ

q

(α + β) . P r o o f. Let

S = {(σ, τ ) ∈ Z

_p

× Z

_p

| p-adic expansions of − σ and − τ satisfy (4.1)}.

Note that if (σ, τ ) ∈ S, then (σ

⁰

, τ

⁰

) ∈ S, so (σ

⁽ⁱ⁾

, τ

⁽ⁱ⁾

) ∈ S for any i.

From Theorem 2 in [3],

Θ(α

⁽ⁱ⁾

, β

⁽ⁱ⁾

; 1) = Γ

_p

(α

⁽ⁱ⁾

)Γ

_p

(β

⁽ⁱ⁾

) Γ

p

(α

⁽ⁱ⁾

+ β

⁽ⁱ⁾

) ,

where Θ(σ, τ ; X) is the analytic element which extends the ratio

2

F

1

 σ, τ 1 ; X



2

F

₁⁽¹⁾

 σ, τ 1 ; X

^p

 .

Then we have

α

⁽ⁱ⁾

+ β

⁽ⁱ⁾

= (α + β)

⁽ⁱ⁾

for any i,

t−1

Y

i=0

Γ

_p

(α

⁽ⁱ⁾

) = Γ

_q

(α),

t−1

Y

i=0

Γ

_p

(β

⁽ⁱ⁾

) = Γ

_q

(β) and

t−1

Y

i=0

Γ

_p

((α + β)

⁽ⁱ⁾

) = Γ

_q

(α + β).

As

2

F

₁^(t)

 α, β 1 ; 1



=

t−1

Y

i=0

Θ(α

⁽ⁱ⁾

, β

⁽ⁱ⁾

; 1),

the result follows.

R e m a r k. Let a = P

_t−1

i=0

a

i

p

ⁱ

and b = P

_t−1

i=0

b

i

p

ⁱ

be p-adic expansions of a and b satisfying 0 < a, b < q − 1 and a + b < q − 1. If a

_i

+ b

_i

< p for any i with 0 ≤ i ≤ t − 1, then

2

F

₁^(t)



_a

q−1

,

_q−1^b

1 ; 1



= J(ω

_q^−a

, ω

_q^−b

) (cf. (3.3) in [9]).

Set α = a/(q − 1) and β = b/(q − 1). Then (α, β) ∈ S and ν

_p ^a+b_b

= 0, so the statement holds from Remark 2 after Proposition 1 and Proposition 4.

In order to use a uniform convergence argument as in [9], we seem to need not only Dwork’s theorem (Theorem 1.1 in [2], Theorem 2.3 in [9]) but also a “truncated” version of it, which is the following:

Theorem 5. For an integer r ≥ 0, let A

^(r)

(m) and g

_r

(m) take values in Z

_p

for non-negative integers m. Moreover , suppose that A

^(r)

and g

_r

satisfy the following:

1. |A

^(r)

(0)| = 1;

2. A

^(r)

(m) ∈ g

r

(m)Z

p

for m ≥ 0;

3. For a positive integer c with p

^L

≤ c < p

^L+1

, L > 0, A

^(r)

(m) 6= 0 and g

_r

(m) 6= 0 for m ≤ h

_r

(c);

4. For integers s and r with 0 ≤ r ≤ L − 1 and 0 ≤ s ≤ L − r − 1, and for integers a, µ and m with 0 ≤ a < p, 0 ≤ µ < p

^s

and m ≥ 0, we have

A

^(r)

(a + pµ + mp

^s+1

)

A

^(r)

(a + pµ) − A

^(r+1)

(µ + mp

^s

)

A

^(r+1)

(µ) ∈ p

^s+1

g

_s+r+1

(m) g

r

(a + pµ) Z

_p

. Then for 0 ≤ s ≤ L − 1 and m ≥ 0, we have

F (X)G

_m,s

(X

^p

) ≡ G(X

^p

)F

_m,s+1

(X) (mod g

_s+1

(m)p

^s+1

[[X]]), where

F (X) = X

∞ n=0

A

⁽⁰⁾

(n)X

ⁿ

, G(X) = X

∞ n=0

A

⁽¹⁾

(n)X

ⁿ

,

F

_m,s

(X) =

(m+1)p

X

^s−1 n=mp^s

A

⁽⁰⁾

(n)X

ⁿ

and G

_m,s

(X) =

(m+1)p

X

^s−1 n=mp^s

A

⁽¹⁾

(n)X

ⁿ

. Notice that under our assumptions on the A

^(r)

, the denominators in hypothesis 4 are non-zero. The proof of Theorem 1.1 in [2] works as well in this case.

Theorem 6. Let θ

1

, . . . , θ

n

and σ

1

, . . . , σ

n−1

be elements of Q ∩ Z

p

, and

suppose that none of the σ

_i

are zero or negative integers, σ

_i

6= 1 for 1 ≤ i

≤ q

⁰

and σ

i

= 1 for i > q

⁰

. Set F (X) =

_n

F

_n−1

 θ

₁

, . . . , θ

_n

σ

₁

, . . . , σ

_n−1

; X



= X

∞ m=0

A(m)X

^m

. Suppose further that

(1) |σ

_j^(ν)

| = 1 for 1 ≤ j ≤ q

⁰

and ν ≥ 0;

(2) for a positive integer c with p

^L

≤ c < p

^L+1

, L > 0,

(6.1) A(m)

 = 0 for m > c, 6= 0 for m ≤ c, i.e., for some i we have θ

_i

= −c;

(3) for each fixed i with 0 ≤ i ≤ L, supposing that the indices are rear- ranged so that µ

⁽ⁱ⁾_θ

1

≤ . . . ≤ µ

⁽ⁱ⁾_θ

n

and µ

⁽ⁱ⁾σ₁

≤ . . . ≤ µ

⁽ⁱ⁾σ_q0

, we have µ

⁽ⁱ⁾σ_j

> µ

⁽ⁱ⁾_θ

for j = 1, . . . , q

⁰

;

j+1

(4) for i with θ

_i

= −c, µ

^(r)σ_l

6= µ

^(r)_θi

for l and r with 1 ≤ l ≤ q

⁰

and 1 ≤ r ≤ L.

Set

(6.2) A

^(s)

(m) = Q

_n

i=1

(θ

^(s)_i

)

m

m! Q

_n−1

j=1

(σ

_j^(s)

)

_m

and g

_s

(m) = A

^(s)

(m) Q

_q0

j=1

(m + σ

^(s)_j

) . Then conditions of Theorem 5 are satisfied and we get, for s and m with 0 ≤ s ≤ L − 1 and m ≥ 0,

F (X)F

_m,s⁰

(X

^p

) ≡ F

⁰

(X

^p

)F

m,s+1

(X) (mod g

s+1

(m)p

^s+1

[[X]]), where

F

⁰

(X) =

n

F

_n−1⁽¹⁾

 θ

1

, . . . , θ

n

σ

₁

, . . . , σ

_n−1

; X

 . Notice that from assumption (2) we have

(6.1

⁰

) A

^(r)

(m)

 = 0 for m > h

_r

(c), 6= 0 for m ≤ h

_r

(c).

P r o o f o f T h e o r e m 6. This is a truncated version of Theorem 3.1 of [2] which was proved by using Lemma 1 of [1], and Lemmas 2.1, 2.2, 3.1 and 3.2 of [2]. Here we only give corresponding statements without proofs except for Lemma A.

For an integer a in {0, 1, . . . , p − 1} and θ in Z

p

, we define a function

% by

%(a, θ) =

n 1 if a > µ

_θ

, 0 otherwise.

Also we use %(a+, x) to denote the limit of %(b, x) as b approaches a from

the right (cf. [2]).

Lemma A ([1], Lemma 1). Let θ be an element of Q ∩ Z

p

and let a, µ, m and s be integers such that 0 ≤ a < p and µ, m, s ≥ 0.

(1) If (θ)

_a+pµ+mp^s+1

6= 0, then (θ)

_a+pµ+mps+1

(θ

⁰

)

µ+mp^s

≡ (θ)

_mps+1

(θ

⁰

)

mp^s

· (θ)

a+pµ

(θ

⁰

)

µ



1 + mp

^s

θ

⁰

+ µ



_%(a,θ)

(mod 1 + p

^s+1

), where we define



1 + mp

^s

θ

⁰

+ µ



_%(a,θ)

= 1 for the case θ

⁰

+ µ = 0, and

(θ)

_mps+1

(θ

⁰

)

_mps

≡ (−p)

^mps

(mod 1 + p

^s+1

).

(2) If (θ)

a+pµ

6= 0, then ν

_p

 (θ)

_a+pµ

(θ

⁰

)

µ



= µ + (1 + ν

_p

(µ + θ

⁰

))%(a, θ),

where we define (1 + ν

p

(µ + θ

⁰

))%(a, θ) = 0 for µ with µ + θ

⁰

= 0. (We show in the proof that when µ + θ

⁰

= 0, %(a, θ) = 0.)

P r o o f. When θ is not zero nor a negative integer, this is Lemma 1 in [1]. For θ = 0, we have a = µ = m = θ

⁰

= 0, so the lemma holds.

For a negative integer θ, the proof of (1) goes exactly in the same way as that of Lemma 1, so we only prove (2). Let θ

⁰

− 1 = P

_∞

s=0

β

_s

p

^s

be the p-adic expansion and α be such that θ −1 ≡ α (mod p), α ∈ {0, 1, . . . , p−1}. Then

θ = 1 + α + X

∞ s=0

β

_s

p

^s+1

. For an integer r ≥ 2, set

θ

_r

= 1 + α + X

r−2 s=0

β

_s

p

^s+1

and θ

⁰_r

= 1 +

r−1

X

s=0

β

_s

p

^s

.

As θ

r

and θ

_r⁰

approach θ and θ

⁰

respectively as r → ∞, and as (θ)

a+pµ

6= 0, there exists N such that for r ≥ N ,

ν

p

((θ

r

)

a+pµ

) = ν

p

((θ)

a+pµ

) and ν

p

((θ

⁰_r

)

µ

) = ν

p

((θ

⁰

)

µ

).

We take N so large that for r ≥ N , β

_s

= p − 1 for any s ≥ r − 1. As in the proof of Lemma 1 of [1], the left side of (2) equals

(∗) = 1

p − 1 {a + α + (p − 1)µ − β

r−1

− s(θ

r

+ a + pµ − 1) + s(θ

_r⁰

+ µ − 1)}, where for a non-negative integer n with the p-adic expansion n = P

_k

i=0

n

i

p

ⁱ

, s(n) = P

_k

i=0

n

_i

.

1. If a + α < p, then %(a, θ) = 0, and

θ

r

+ a + pµ − 1 = a + α + pA, θ

⁰_r

+ µ − 1 = A + β

r−1

p

^r−1

, where A = P

_r−2

s=0

β

_s

p

^s

+ µ.

Suppose that A ≥ p

^r−1

. Then for r ≥ N , θ = 1 + α +

r−2

X

s=0

β

_s

p

^s+1

− p

^r

and

θ + (a + pµ − 1) = a + α + pA − p

^r

≥ a + α ≥ 0.

This means that (θ)

_a+pµ

= 0, which contradicts our assumption. Hence A < p

^r−1

and we have

s(θ

_r

+ a + pµ − 1) = a + α + s(A) and s(θ

_r⁰

+ µ − 1) = s(A) + β

_r−1

, so (∗) = µ as desired.

2. If a + α ≥ p then %(a, θ) = 1, and

θ

_r

+ a + pµ − 1 = (a + α − p) + pB, θ

⁰_r

+ µ = B + β

_r−1

p

^r−1

, where B = µ + 1 + P

_r−2

s=0

β

s

p

^s

= A + 1.

Suppose that B ≥ p

^r−1

. Then as before we get a contradiction. So B <

p

^r−1

and

s(θ

_r

+ a + pµ − 1) = a + α − p + s(B) and s(θ

⁰_r

+ µ) = s(B) + β

_r−1

. If µ + θ

⁰

6= 0, then the same proof as for Lemma 1 of [1] works. If µ + θ

⁰

= 0, then

θ + a + pµ − 1 = pθ

⁰

− µ

_θ

+ a + pµ − 1 = a − µ

_θ

− 1 < 0, since (θ)

_a+pµ

6= 0, hence %(a, θ) = 0.

For Lemmas B through E, let θ

₁

, . . . , θ

_n

and σ

₁

, . . . , σ

_n−1

be elements of Q ∩ Z

p

, and suppose that none of the σ

i

are zero or negative integers, σ

i

6= 1 for 1 ≤ i ≤ q

⁰

and σ

_i

= 1 for i > q

⁰

. Let g

_s

be defined by (6.2). We put for ν ≥ 0,

N

_θ(ν)

(a) = X

n i=1

%(a, θ

^(ν)_i

), N

_θ(ν)

(a+) = X

n i=1

%(a+, θ

_i^(ν)

),

N

_σ(ν)

(a) =

q⁰

X

j=1

%(a, σ

^(ν)_j

), N

_σ(ν)

(a+) =

q⁰

X

j=1

%(a+, σ

^(ν)_j

).

In particular, we put

N

_θ

(a) = N

_θ(0)

(a) and N

_σ

(a+) = N

_σ(0)

(a+).