Connection Formula of Basic Hypergeometric Series rϕr−1(0; b; q, x)

 J. Math. Tokushima Univ.

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Connection Formula of Basic Hypergeometric

Series

ϕ

(0; b; q, x)

By

Yousuke Ohyama

Department of Mathematical Sciences, Graduate School of Science and Technology Tokushima University, Tokushima 770-8506, JAPAN

e-mail address : [email protected]

(Received October 13, 2017)

Abstract

We show a connection formula of a linear q-differential equation satisfied by rϕr−1(0; b; q, x) where any element of b are not zero.

We use a q-Laplace transformation to obtain an integral represen-tation of solutions of the q-differential equation.

2010 Mathematics Subject Classification. Primary 34M40;Secondary 33D15

1

Introduction

We show a connection formula of a linear q-differential equation satisfied by

rϕr−1(0, 0, ..., 0; b1, ..., br−1; q, x) in case that b1b2· · · br−1̸= 0. The basic

hyper-geometric seriesrϕr−1(0, 0, ..., 0; b1, ..., br−1; q, x) satisfies a linear q-differential

equation of the r-th order: [ x− (1 − σq) r_∏−1 k=1 (1−b_qkσq) ] y(x) = 0, (1)

where σqy(x) = y(xq). The condition b1b2· · · br−1 ̸= 0 implies that the origin

is a regular singular point of (1). Around the infinity (1) has r solutions which are represented by convergent power series on x1/r_{. In this sense, (1) is the}

most degenerate case of hypergeometric equations.

Thomae [6, 7] showed a connection formula on 2ϕ1(a1, a2; b1) and 3ϕ2(a1, a2, a3; b1, b2). In [8] Watson gave connection formulae in more general cases. He showed a connection formula of rϕr−1(a1, a2, .., ar; b1, ..., bs, 0; q, z), where s <

r. Watson also showed that an asymptotic expansion ofs+1ϕr−1(a′1, a′2, ..., a′s+1;

b′1, ..., b′r−1; q, z) (he used a notations+1Pr−1), but he did not give a

resumma-tion of divergent series. Later Slater [4, 5] also gave a more general form of a connection formula.

J.-P. Ramis, J. Sauloy and C. Zhang started modern study on divergent

q-series and a q-analogue of the Stokes phenomenon [3]. Zhang studied the q-Stokes phenomenon of q-confluent hypergeometric function 2ϕ0(a, b; 0; q, x) [10]. He has also shown a connection formula of Jackson’s q-analogue of the Bessel function Jν(1) [11]. Since

Jν(1)(x; q) = (qν+1_{; q)} ∞ (q; q)∞ ( x 2 )ν 2ϕ1 ( 0, 0; qν+1; q,−x 2 4 ) ,

the connection formula of Jν(1)(x; q) is essentially the case r = 2 of (1).

Since all of local solutions around the origin and the infinity are repre-sented by convergent power series, we can determine the connection formula by a q-Laplace transformation [9]. We show a useful formula on p-Laplace transformation applied to q-difference equations (pm_{= q) in section two.}

We show a connection formula in section three. We study the q-differential

equation _[ xr r ∏ k=1 (1_{− a}kσp)− ( −σ_ppr )r] u(x) = 0.

Local solutions around the infinity are

u1,_∞(x) = θp(−a1x)

θp(−x) r

ϕr−1

(

0, 0, ..., 0

pr_{a1/a2, p}r_{a2/a3, ..., p}r_a1/a r; p

r_, 1

a1a2· · · arxr

)

and u2,∞(x), ..., ur,∞(x) are obtained by the cyclic transformation of a1, a2, ..., ar.

We take a a primitive r-th root ω of unity. Local solutions around the origin are uj,0(x) = 1 θp(−ωjp(1−r)/2x) vj(x), vj(x) = ∞ ∑ n=0 v(j)n xn,

for j = 0, 1, 2, ..., r _{− 1. We assume that v}(j)₀ = 1. The connection formula between (u0,0, u1,0, ..., ur−1,0) and (u1,∞, ..., ur,∞) is given by

vj(x) = 1 (q, a2/a1, ..., ar/a1; q)∞ θp(−ωjp(1−r)/2a1x)θp(−x) θp(−ωjp(1−r)/2x)θp(−a1x) u1,∞(x) + idem(a1; a2, ..., ar).

The symbol ”idem (a1; a2, ..., ar)” stands for the sum of the r expressions

ob-tained from the preceding expression by interchanging a1with each a2, a3, ..., ar.

The author gives his gratitude to Professor Changgui Zhang for fruitful dis-cussions. Some works has done during his stay at Lille on September 2017. This work is supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) Number 6K05176.

2

Notations

We denote the m-vetcor (0, 0, ..., 0) by 0m.

We assume that 0 <|q| < 1. For n = 0, 1, 2, ..., we set the q-shifted factorial

(a; q)n= n_∏−1 j=0 (1− aqj_{), (a; q)} ∞= ∞ ∏ j=0 (1− aqj_).

We set (a1, a2, ..., am; q)n=∏mj=1(aj; q)n for n = 0, 1, 2, ... or n =∞.

We set the theta function

θq(x) := θ(x) =

∑

k∈Z

qk(k−1)/2xk= (q,−x, −q/x; q)∞.

The theta function satisfies

θq(qkx) = q−k(k−1)/2x−kθq(x) (k∈ Z),

xθq(1/x) = θq(x), θq(1/x) = θq(qx).

The basic hypergeometric series [1] is defined by

rϕs(a1, . . . , ar; b1, . . . , bs; q, x) :=∑ n≥0 (a1, . . . , ar; q)n (b1, . . . , bs; q)n(q; q)n { (₋₁₎nqn(n2−1) }1+s−r xn.

Let σq be a q-shift operator σq[f (x)] = f (xq). When 1 + s ≥ r, rϕs is

convergent and satisfies a q-difference equation with (s + 1)-th order  x r ∏ j=1 (1− ajσq)− (1 − σq) r_∏−1 k=1 (1−b_qkσq)   y(x) = 0.

3

q-Borel transformation and q-Laplace

trans-formation

We review a q-Borel transformation and a q-Laplace transformation. See [9, 3] for detail.

Connection Formula of Basic Hypergeometric Series rφr-1(0; b; q, x) 

Thomae [6, 7] showed a connection formula on 2ϕ1(a1, a2; b1) and3ϕ2(a1, a2, a3; b1, b2). In [8] Watson gave connection formulae in more general cases. He showed a connection formula ofrϕr−1(a1, a2, .., ar; b1, ..., bs, 0; q, z), where s <

r. Watson also showed that an asymptotic expansion ofs+1ϕr−1(a′1, a′2, ..., a′s+1;

b′1, ..., b′r−1; q, z) (he used a notations+1Pr−1), but he did not give a

resumma-tion of divergent series. Later Slater [4, 5] also gave a more general form of a connection formula.

J.-P. Ramis, J. Sauloy and C. Zhang started modern study on divergent

q-series and a q-analogue of the Stokes phenomenon [3]. Zhang studied the q-Stokes phenomenon of q-confluent hypergeometric function 2ϕ0(a, b; 0; q, x) [10]. He has also shown a connection formula of Jackson’s q-analogue of the Bessel function Jν(1) [11]. Since

Jν(1)(x; q) = (qν+1_{; q)} ∞ (q; q)∞ ( x 2 )ν 2ϕ1 ( 0, 0; qν+1; q,−x 2 4 ) ,

the connection formula of Jν(1)(x; q) is essentially the case r = 2 of (1).

Since all of local solutions around the origin and the infinity are repre-sented by convergent power series, we can determine the connection formula by a q-Laplace transformation [9]. We show a useful formula on p-Laplace transformation applied to q-difference equations (pm_{= q) in section two.}

We show a connection formula in section three. We study the q-differential

equation _[ xr r ∏ k=1 (1_{− a}kσp)− ( −σ_ppr )r] u(x) = 0.

Local solutions around the infinity are

u1,_∞(x) =θp(−a1x)

θp(−x) r

ϕr−1

(

0, 0, ..., 0

pr_{a1/a2, p}r_{a2/a3, ..., p}r_a1/a r; p

r_, 1

a1a2· · · arxr

)

and u2,∞(x), ..., ur,∞(x) are obtained by the cyclic transformation of a1, a2, ..., ar.

We take a a primitive r-th root ω of unity. Local solutions around the origin are uj,0(x) = 1 θp(−ωjp(1−r)/2x) vj(x), vj(x) = ∞ ∑ n=0 v(j)n xn,

for j = 0, 1, 2, ..., r_{− 1. We assume that v0}(j) = 1. The connection formula between (u0,0, u1,0, ..., ur−1,0) and (u1,∞, ..., ur,∞) is given by

vj(x) = 1 (q, a2/a1, ..., ar/a1; q)∞ θp(−ωjp(1−r)/2a1x)θp(−x) θp(−ωjp(1−r)/2x)θp(−a1x) u1,∞(x) + idem(a1; a2, ..., ar).

The symbol ”idem (a1; a2, ..., ar)” stands for the sum of the r expressions

ob-tained from the preceding expression by interchanging a1with each a2, a3, ..., ar.

The author gives his gratitude to Professor Changgui Zhang for fruitful dis-cussions. Some works has done during his stay at Lille on September 2017. This work is supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) Number 6K05176.

2

Notations

We denote the m-vetcor (0, 0, ..., 0) by 0m.

We assume that 0 <|q| < 1. For n = 0, 1, 2, ..., we set the q-shifted factorial

(a; q)n= n_∏−1 j=0 (1− aqj_{), (a; q)} ∞= ∞ ∏ j=0 (1− aqj_).

We set (a1, a2, ..., am; q)n=∏mj=1(aj; q)n for n = 0, 1, 2, ... or n =∞.

We set the theta function

θq(x) := θ(x) =

∑

k∈Z

qk(k−1)/2xk= (q,−x, −q/x; q)∞.

The theta function satisfies

θq(qkx) = q−k(k−1)/2x−kθq(x) (k∈ Z),

xθq(1/x) = θq(x), θq(1/x) = θq(qx).

The basic hypergeometric series [1] is defined by

rϕs(a1, . . . , ar; b1, . . . , bs; q, x) :=∑ n≥0 (a1, . . . , ar; q)n (b1, . . . , bs; q)n(q; q)n { (₋₁₎nqn(n2−1) }1+s−r xn.

Let σq be a q-shift operator σq[f (x)] = f (xq). When 1 + s ≥ r, rϕs is

convergent and satisfies a q-difference equation with (s + 1)-th order  x r ∏ j=1 (1− ajσq)− (1 − σq) r_∏−1 k=1 (1−b_qkσq)   y(x) = 0.

3

q-Borel transformation and q-Laplace

trans-formation

We review a q-Borel transformation and a q-Laplace transformation. See [9, 3] for detail.

The q-Borel transformation_B− q :C[[x]] → C[[τ]] is defined by B− q [_∞ ∑ n=0 anxn ] := ∞ ∑ n=0 anq−n(n−1)/2τn.

We identify a germ of holomorphic functions at the origin_OC,0 as a subset of C[[x]]. As a linear operator on C[[x]], the following lemma is useful to study

q-difference equations.

Lemma 1. (1) The q-Borel transformation B−

q shifts the power of σq:

B−q(xmσnqf ) = q−m(m−1)/2τmσqn−mB−q(f ).

(2) Multiplication by the theta function shifts the power of x: xmσnq [ 1 θq(cx) f (x) ] =q n(n−1)/2_cn θq(cx) xm+nσqnf (x).

The inverse transformation of _B−

q is given by a q-Laplace transform L−q.

Assume that φ(τ ) is holomorphic on_{|τ| ≦ ε. We define}

L−qφ(x) = 1 2πi ∫ |τ|=ε φ(τ )θq(x/τ ) dτ τ .

Under a suitable condition, we have_L−

q ◦ Bq−f = f .

We consider the p-Laplace transform of a ratio of pm_-products.

Proposition 2. Let m be a positive integers. We set pm_{= q. We assume that}

s + m≦ r. When s + m = r, we need |q(1+m)/2_b1

· · · bs/a1a2· · · arxm| < 1. We

consider the contour integral I = 1 2πi ∫ |τ|=ε ∏s j=1(bjτ ; q)∞ ∏r k=1(akτ ; q)∞ θp(x/τ ) dτ τ ,

where ∏r_k=1(akτ ; q)∞ does not have any zero on|τ| ≦ ε. Then we obtain

I = (b1/a1, ..., bs/a1; q)∞ (q, a2/a1, ..., ar/a1; q)∞ θp(a1x) ×s+mϕr−1 ( qa1/b1, ..., qa1/bs, 0m qa1/a2, ..., qa1/ar ; q, (₋₁₎r_qr−s+(1−m)/2_{b1· · · b} s am+s1 −r+1a2· · · arxm ) + idem(a1; a2, ..., ar). (2)

Proof. The following relations are directly proved :

Resτ =1/aqn 1 (aτ ; q)_∞ dτ τ =− (₋₁₎n_qn(n+1)/2 (q; q)_∞(q; q)n ,

θp(x/τ )|τ→1/aqn = (ax)−mnp−nm(nm−1)/2θp(ax),

(bτ ; q)∞|τ→1/aqn= (−b/a)nq−n(n+1)/2(b/a; q)_∞(aq/b; q)n.

By using the above relations we can show Proposition.

4

Connection formula

We consider the equation  z r ∏ j=k (1− akσq)− ( −σ_qq )r  y(z) = 0. (3)

Local solutions of (3) around the infinity are

y1,_∞(z) = θq(−a1z)

θq(−z) r

ϕr−1

(

0, 0, ..., 0

qa1/a2, qaj/a3, ..., qa1/ar; q,

1

a1· · · arz

)

and y2,∞(z), ..., yr,∞(z) are obtained by the cyclic transform a1→ a2→ · · · → ar→ a1.

Since (3) has ramified solutions around the origin, we take a covering trans-formation z = xr_{. We set p}r_{= q.} [ xr r ∏ k=1 (1− akσp)− ( −σ_qp )r] u(x) = 0. (4)

We give a connection formula of (4). Local solutions of (4) around the infinity are

u1,∞(x) = θp_θ(−a1x) p(−x) rϕr−1

(

0, 0, ..., 0

qa1/a2, qa2/a3, ..., qa1/ar; q,

1

a1a2_{· · · a}rxr

)

and u2,∞(x), ..., ur,∞(x) are obtained by the cyclic transform.

We take a complex number ω, which is a primitive r-th root of unity:

ωr_{= 1. Local solutions of (4) around the origin are}

uj,0(x) = 1 θp(−ωjp(1−r)/2x) vj(x), vj(x) = ∞ ∑ n=0 v(j) n xn,

for j = 0, 1, 2, ..., r_{−1. We assume that v0}(j)= 1. We show a connection formula between (u0,0, u1,0, ..., ur−1,0) and (u1,∞, ..., ur,∞).

We set elementary symmetric polynomials s1, s2, ..., srso that r ∏ k=1 (1− akx) = r ∑ k=1 (−1)k_s kxk.

Connection Formula of Basic Hypergeometric Series rφr-1(0; b; q, x) 33

The q-Borel transformation_B−

q :C[[x]] → C[[τ]] is defined by B− q [_∞ ∑ n=0 anxn ] := ∞ ∑ n=0 anq−n(n−1)/2τn.

We identify a germ of holomorphic functions at the origin_OC,0 as a subset of C[[x]]. As a linear operator on C[[x]], the following lemma is useful to study

q-difference equations.

Lemma 1. (1) The q-Borel transformationB−

q shifts the power of σq:

B−q (xmσnqf ) = q−m(m−1)/2τmσqn−mB−q(f ).

(2) Multiplication by the theta function shifts the power of x: xmσnq [ 1 θq(cx) f (x) ] =q n(n−1)/2_cn θq(cx) xm+nσqnf (x).

The inverse transformation of _B−

q is given by a q-Laplace transform L−q.

Assume that φ(τ ) is holomorphic on_{|τ| ≦ ε. We define}

L−q φ(x) = 1 2πi ∫ |τ|=ε φ(τ )θq(x/τ ) dτ τ .

Under a suitable condition, we have_L−

q ◦ Bq−f = f .

We consider the p-Laplace transform of a ratio of pm_-products.

Proposition 2. Let m be a positive integers. We set pm_{= q. We assume that}

s + m≦ r. When s + m = r, we need |q(1+m)/2_b1

· · · bs/a1a2· · · arxm| < 1. We

consider the contour integral I = 1 2πi ∫ |τ|=ε ∏s j=1(bjτ ; q)∞ ∏r k=1(akτ ; q)∞ θp(x/τ ) dτ τ ,

where∏r_k=1(akτ ; q)∞ does not have any zero on|τ| ≦ ε. Then we obtain

I = (b1/a1, ..., bs/a1; q)∞ (q, a2/a1, ..., ar/a1; q)∞ θp(a1x) ×s+mϕr−1 ( qa1/b1, ..., qa1/bs, 0m qa1/a2, ..., qa1/ar ; q, (₋₁₎r_qr−s+(1−m)/2_{b1· · · b} s am+s1 −r+1a2· · · arxm ) + idem(a1; a2, ..., ar). (2)

Proof. The following relations are directly proved :

Resτ =1/aqn 1 (aτ ; q)_∞ dτ τ =− (₋₁₎n_qn(n+1)/2 (q; q)_∞(q; q)n ,

θp(x/τ )|τ→1/aqn= (ax)−mnp−nm(nm−1)/2θp(ax),

(bτ ; q)∞|τ→1/aqn= (−b/a)nq−n(n+1)/2(b/a; q)_∞(aq/b; q)n.

By using the above relations we can show Proposition.

4

Connection formula

We consider the equation  z r ∏ j=k (1− akσq)− ( −σ_qq )r  y(z) = 0. (3)

Local solutions of (3) around the infinity are

y1,_∞(z) = θq(−a1z)

θq(−z) r

ϕr−1

(

0, 0, ..., 0

qa1/a2, qaj/a3, ..., qa1/ar; q,

1

a1· · · arz

)

and y2,∞(z), ..., yr,∞(z) are obtained by the cyclic transform a1→ a2→ · · · → ar→ a1.

Since (3) has ramified solutions around the origin, we take a covering trans-formation z = xr_{. We set p}r_{= q.} [ xr r ∏ k=1 (1− akσp)− ( −σ_qp )r] u(x) = 0. (4)

We give a connection formula of (4). Local solutions of (4) around the infinity are

u1,∞(x) = θp_θ(−a1x) p(−x) rϕr−1

(

0, 0, ..., 0

qa1/a2, qa2/a3, ..., qa1/ar; q,

1

a1a2_{· · · a}rxr

)

and u2,∞(x), ..., ur,∞(x) are obtained by the cyclic transform.

We take a complex number ω, which is a primitive r-th root of unity:

ωr_{= 1. Local solutions of (4) around the origin are}

uj,0(x) = 1 θp(−ωjp(1−r)/2x) vj(x), vj(x) = ∞ ∑ n=0 v(j) n xn,

for j = 0, 1, 2, ..., r_{−1. We assume that v0}(j)= 1. We show a connection formula between (u0,0, u1,0, ..., ur−1,0) and (u1,∞, ..., ur,∞).

We set elementary symmetric polynomials s1, s2, ..., sr so that r ∏ k=1 (1− akx) = r ∑ k=1 (−1)k_s kxk.

We set cj=−ωjp(1−r)/2. Then vj(x) satisfies a q-difference equation [_∑r k=1 (−1)k_ck jpk(k−1)/2skxkσpk− σrp ] vj(x) = 0. We remark that σr p= σq. Since wj(τ ) = (B−pvj)(τ ) satisfies [ _r ∏ k=1 (1_{− c}jakτ )− σrp ] wj(τ ) = 0, we have wj(τ ) = 1 (cja1τ, cja2τ, ..., cjarτ ; q)∞ . By (2) we obtain vj(x) =L−pwj(x) = 1 2πi ∫ |τ|=ε 1 (cja1τ, cja2τ, ..., cjarτ ; q)∞ θp(x/τ ) dτ τ = θp(cja1x) (q, a2/a1, ..., ar/a1; q)∞rϕr−1 ( 0, 0, ..., 0 qa1/a2, ..., qa1/ar; q, (−1)r_q(1−r)/2 cr ja1a2· · · arxr ) + idem(a1; a2, ..., ar) = θp(cja1x) (q, a2/a1, ..., ar/a1; q)∞rϕr−1 ( 0, 0, ..., 0 qa1/a2, ..., qa1/ar; q, 1 a1a2_{· · · a}rxr ) + idem(a1; a2, ..., ar). We remark that cr j = (−1)rq(1−r)/2.

The main result is as follows:

Theorem 3. We take a primitive r-th root ω of unity. A connection formula

of (4) is given by uj,0(x) = 1 (q, a2/a1, ..., ar/a1; q)∞ θp(−ωjp(1−r)/2a1x)θp(−x) θp(−ωjp(1−r)/2x)θp(−a1x) u1,∞(x) + idem(a1; a2, ..., ar), for j = 0, 1, ..., r_{− 1.} The case r = 2:

We set p2_{= q. We take a p-difference equation}

[

p2x2(1− a1σp)(1− a2σp)− σp2

]

u(x) = 0. (5)

We give a connection formula of (5). Local solutions of (5) around the infinity are u1,∞(x) = θp_θ(−a1x) p(−x) 2ϕ1 ( 0, 0; qa1/a2; q, 1 a1a2x2 ) , u2,∞(x) = θp_θ(−a2x) p(−x) 2ϕ1 ( 0, 0; qa2/a1; q, 1 a1a2x2 ) .

Local solutions of (5) around the origin are

u0,1(x) = 1 θp(−p−1/2x) v1(x), v1(x) = ∞ ∑ n=0 v(1)n xn, u0,2(x) = 1 θp(p−1/2x) v2(x), v2(x) = ∞ ∑ n=0 v(2) n xn.

We assume that v₀(j)= 1 for j = 1, 2. By Theorem 3 we obtain

u0,1(x) = 1 (q, a2/a1; q)∞ θp(−p1/2a1x)θp(−x) θp(−p1/2x)θp(−a1x) u1,∞(x) + 1 (q, a1/a2; q)_∞ θp(−p1/2a2x)θp(−x) θp(−p1/2x)θp(−a2x) u2,∞(x), u0,2(x) = 1 (q, a2/a1; q)∞ θp(p1/2a1x)θp(−x) θp(p1/2x)θp(−a1x) u1,_∞(x) + 1 (q, a1/a2; q)∞ θp(−p1/2a2x)θp(−x) θp(−p1/2x)θp(−a2x) u2,_∞(x).

This connection formula is essentially equivalent to the connection formula of Jackson’s first q-Bessel functions in [11].

5

Conclusion

We show a connection formula of (4), which is a generalization of Jackson’s first

q-analogue of the Bessel functions [2]. We can obtain a connection formula of

solutions represented by a convergent (non-hypergeometric) series of x1/m _by

applying the p-Laplace transformation (2) to a product of pm_{-shifted factorials}

p for other q-hypergeometric equations.

We should study the q-Stokes phenomenon [3] for divergent series solutions. By using the other q-Borel transformationB+

q, we can give a resummation for

Connection Formula of Basic Hypergeometric Series rφr-1(0; b; q, x) 35

We set cj =−ωjp(1−r)/2. Then vj(x) satisfies a q-difference equation

[_∑r k=1 (−1)k_ck jpk(k−1)/2skxkσpk− σrp ] vj(x) = 0. We remark that σr p = σq. Since wj(τ ) = (B−pvj)(τ ) satisfies [ _r ∏ k=1 (1_{− c}jakτ )− σrp ] wj(τ ) = 0, we have wj(τ ) = 1 (cja1τ, cja2τ, ..., cjarτ ; q)∞ . By (2) we obtain vj(x) =L−pwj(x) = 1 2πi ∫ |τ|=ε 1 (cja1τ, cja2τ, ..., cjarτ ; q)∞ θp(x/τ ) dτ τ = θp(cja1x) (q, a2/a1, ..., ar/a1; q)∞rϕr−1 ( 0, 0, ..., 0 qa1/a2, ..., qa1/ar; q, (−1)r_q(1−r)/2 cr ja1a2· · · arxr ) + idem(a1; a2, ..., ar) = θp(cja1x) (q, a2/a1, ..., ar/a1; q)∞rϕr−1 ( 0, 0, ..., 0 qa1/a2, ..., qa1/ar; q, 1 a1a2_{· · · a}rxr ) + idem(a1; a2, ..., ar). We remark that cr j = (−1)rq(1−r)/2.

The main result is as follows:

Theorem 3. We take a primitive r-th root ω of unity. A connection formula

of (4) is given by uj,0(x) = 1 (q, a2/a1, ..., ar/a1; q)∞ θp(−ωjp(1−r)/2a1x)θp(−x) θp(−ωjp(1−r)/2x)θp(−a1x) u1,∞(x) + idem(a1; a2, ..., ar), for j = 0, 1, ..., r_{− 1.} The case r = 2:

We set p2_{= q. We take a p-difference equation}

[

p2x2(1− a1σp)(1− a2σp)− σp2

]

u(x) = 0. (5)

We give a connection formula of (5). Local solutions of (5) around the infinity are u1,∞(x) = θp_θ(−a1x) p(−x) 2ϕ1 ( 0, 0; qa1/a2; q, 1 a1a2x2 ) , u2,∞(x) = θp_θ(−a2x) p(−x) 2ϕ1 ( 0, 0; qa2/a1; q, 1 a1a2x2 ) .

Local solutions of (5) around the origin are

u0,1(x) = 1 θp(−p−1/2x) v1(x), v1(x) = ∞ ∑ n=0 v(1)n xn, u0,2(x) = 1 θp(p−1/2x) v2(x), v2(x) = ∞ ∑ n=0 v(2) n xn.

We assume that v₀(j)= 1 for j = 1, 2. By Theorem 3 we obtain

u0,1(x) = 1 (q, a2/a1; q)∞ θp(−p1/2a1x)θp(−x) θp(−p1/2x)θp(−a1x) u1,∞(x) + 1 (q, a1/a2; q)_∞ θp(−p1/2a2x)θp(−x) θp(−p1/2x)θp(−a2x) u2,∞(x), u0,2(x) = 1 (q, a2/a1; q)∞ θp(p1/2a1x)θp(−x) θp(p1/2x)θp(−a1x) u1,_∞(x) + 1 (q, a1/a2; q)∞ θp(−p1/2a2x)θp(−x) θp(−p1/2x)θp(−a2x) u2,_∞(x).

This connection formula is essentially equivalent to the connection formula of Jackson’s first q-Bessel functions in [11].

5

Conclusion

We show a connection formula of (4), which is a generalization of Jackson’s first

q-analogue of the Bessel functions [2]. We can obtain a connection formula of

solutions represented by a convergent (non-hypergeometric) series of x1/m _by

applying the p-Laplace transformation (2) to a product of pm_{-shifted factorials}

p for other q-hypergeometric equations.

We should study the q-Stokes phenomenon [3] for divergent series solutions. By using the other q-Borel transformationB+

q, we can give a resummation for

References

[1] Gasper, G., Rahman, M.; Basic Hypergeometric Series, 2nd ed, Cambridge (2004).

[2] Jackson, F. H.; On generalized functions of Legendre and Bessel, Trans.

Roy. Soc. Edinburgh, 41 (1903), 1–28.

[3] Ramis, J.-P., Sauloy, J., Zhang, C.; Local Analytic Classification of q-difference Equations, Ast´erisque 355 (2013).

[4] Slater, L. J.; An integral of hypergeometric type, Proc. Cambridge Philos.

Soc. 48 (1952), 578–582.

[5] Slater, L. J.; Generalized Hypergeometric Functions, Cambridge Univer-sity Press, Cambridge (1966).

[6] Thomae, J.; Beitr¨age zur Theorie der durch die Heinesche Reihe..., J. reine

angew. Math. 70 (1869), 258–281.

[7] Thomae, J.; Les s´eries Hein´eennes sup´erieures, ou les s´eries de la forme ...,

Ann. Mat. Pura Appl. 4, (1870) 105–138.

[8] Watson, G. N.; The continuation of functions defined by generalized hy-pergeometric series, Trans. Camb. Phil. Soc. 21 (1910), 281–299.

[9] Zhang C.; D´eveloppements asymptotiques q-Gevrey et s´eries Gq-sommables Ann. Inst. Fourier 49 (1999), 227–261.

[10] Zhang C.; Une sommation discr`e pour des ´equations aux q-diff´erences lin´eaires et `a coefficients, analytiques: th´eorie g´en´erale et exemples, in “Differential Equations and Stokes Phenomenon”, World Sci. Publ., (2002), 309–329.

[11] Zhang C.; Sur les fonctions q-Bessel de Jackson, J. Approx. Theory 122 (2003), 208–223.

8

Asymptotic Behavior of Solutions for

Kirchhoff Type Dissipative Wave Equations

in Unbounded Domains

Kosuke Ono

Department of Mathematical Sciences, Tokushima University, Tokushima 770-8502, JAPAN

e-mail : [email protected]

(Received September 30, 2017) Abstract

Consider the Cauchy problem for the non-degenerate Kirchhoff type dissipative wave equations with the initial data belonging to

H2_(RN₎

× H1_(RN_{) in unbounded domains. When the coefficient}

ρ or the initial energy E(0) is small at least, we show the global

existence theorem and derive decay estimates of energies in the L2

-frame. Moreover, when the initial data belong to L1_(RN_)×L1_(RN₎

in addition, we improve the decay rates of the solutions. 2010 Mathematics Subject Classification. 35B40, 35L15

1

Introduction

In this paper we consider the Cauchy problem for the non-degenerate Kirch-hoff type dissipative wave equations :

   ρu′′+ ( 1 + ∫ RN|A 1/2_u( ·, t)|2_dx )γ Au + u′= 0 in _RN × [0, ∞) , u(x, 0) = u0(x) and u′(x, 0) = u1(x) in RN, (1.1) where u = u(x, t) is an unknown real value function, ′ _{= ∂/∂t, A = −∆ =}

−∑Nj=1∂2/∂x2j is the Laplace operator with domainD(A) = H2(RN), ρ > 0

and γ > 0 are positive constants.

Equations (1.1) describes small amplitude vibrations of an elastic string when the dimension N is one (see Kirchhoff [9] for the original equation, and also see Carrier [5], Dickey [6]). Equations including non-local terms like (1.1) are called Kirchhoff type.