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Non-terminating q-Whippleの変換公式

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Watsonによる8ϕ7q-Whippleの変換公式
8ϕ7[a,aq,aq,b,c,d,e,qna,a,aq/b,aq/c,aq/d,aq/e,aqn+1;a2qn+2bcde]=(aq,aq/de)n(aq/d,aq/e;q)n4ϕ3[aq/bc,d,e,qnaq/b,aq/c,deqn/a;q]
の一般的となる, non-terminating q-Whippleの変換公式を示す.

8ϕ7[a,aq,aq,b,c,d,e,fa,a,aq/b,aq/c,aq/d,aq/e,aq/f;a2q2bcdef]=(aq,aq/de,aq/df,aq/ef;q)(aq/d,aq/e,aq/f,aq/def;q)4ϕ3[aq/bc,d,e,faq/b,aq/c,def/a;q]+(aq,aq/bc,d,e,f,a2q2/bdef,a2q2/cdef;q)(aq/b,aq/c,aq/d,aq/e,aq/f,a2q2/bcdef,def/aq;q)4ϕ3[aq/de,aq/df,aq/ef,a2q2/bcdefa2q2/bdef,a2q2/cdef,aq2/def;q]

w=a2q/bcd,a3qn+2=bcdefgとする. Baileyのterminating10ϕ9変換公式
10ϕ9[a,aq,aq,b,c,d,e,f,g,qna,a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aqn+1;q]=(aq,aq/ef,wq/e,wq/f;q)n(aq/e,aq/f,wq,wq/ef;q)n10ϕ9[w,wq,wq,wb/a,wc/a,wd/a,e,f,g,qnw,w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1;q]
において, a,c,d,e,f,gを固定してnとする. b=a3qn+2/cdefg,w=efgqn1/aとなるから,
8ϕ7[a,aq,aq,c,d,e,f,ga,a,aq/c,aq/d,aq/e,aq/f,aq/g;a2q2cdefg]=limn(aq,aq/ef,fgqn/a,egqn/a;q)n(aq/e,aq/f,gqn/a,efgqn/a)n10ϕ9[w,wq,wq,wb/a,wc/a,wd/a,e,f,g,qnw,w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1;q]=(aq,aq/ef,aq/eg,aq/fg;q)(aq/e,aq/f,aq/g,aq/efg;q)limn10ϕ9[w,wq,wq,wb/a,wc/a,wd/a,e,f,g,qnw,w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1;q]
nを奇数として, n=2m+1とすると,
=limn10ϕ9[w,wq,wq,wb/a,wc/a,wd/a,e,f,g,qnw,w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1;q]=limnk=0m(w,wq,wq,wb/a,wc/a,wd/a,e,f,g,qn;q)k(w,w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)kqk+limnk=0m(w,wq,wq,wb/a,wc/a,wd/a,e,f,g,qn;q)nk(w,w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)nkqnk
最初の項は,
limnk=0m(w,wq,wq,wb/a,wc/a,wd/a,e,f,g,qn;q)k(w,w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)kqk=limnk=0m(efgqn1/a,wq,wq,aq/cd,cefgqn1/a2,defgqn1/a2,e,f,g,qn;q)k(w,w,cdefgqn1/a2,aq/c,aq/d,fgqn/a,egqn/a,efqn/a,efg/a,q;q)kqk=k=0(aq/cd,e,f,g;q)k(aq/c,aq/d,efg/a,q;q)kqk
であり, 次の項は,
limnk=0m(w,wq,wq,wb/a,wc/a,wd/a,e,f,g,qn;q)nk(w,w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)nkqnklimn(w,wq,wq,wb/a,wc/a,wd/a,e,f,g,qn;q)n(w,w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)nqnk=0m(q1n/w,q1n/w,bqn/a,cqn/a,dqn/a,eqn/w,fqn/w,gqn/w,q2n/w,qn;q)k(q1n/w,qn/w,qn/w,aq1n/wb,aq1n/wc,aq1n/wd,q1n/e,q1n/f,q1n/g,q;q)kqk
係数の部分は,
limn(efgqn1/a,wq,wq,aq/cd,cefgqn1/a2,defgqn1/a2,e,f,g,qn;q)n(w,w,cdefgqn1/a2,aq/c,aq/d,fgqn/a,egqn/a,efqn/a,efg/a,q;q)nqn=limn1efgqn1/aqnefg/aq(aq2/efg,aq/cd,a2q2/cefg,a2q2/defg,e,f,g;q)n(a2q2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/a;q)n=aqefg(aq2/efg,aq/cd,a2q2/cefg,a2q2/defg,e,f,g;q)(a2q2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/a;q)=(aq/efg,aq/cd,a2q2/cefg,a2q2/defg,e,f,g;q)(a2q2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/aq;q)
であり,
limnk=0m(q1n/w,q1n/w,bqn/a,cqn/a,dqn/a,eqn/w,fqn/w,gqn/w,q2n/w,qn;q)k(q1n/w,qn/w,qn/w,aq1n/wb,aq1n/wc,aq1n/wd,q1n/e,q1n/f,q1n/g,q;q)kqk=limnk=0m(q1n/w,q1n/w,a2q2/cdefg,cqn/a,dqn/a,aq/fg,aq/eg,aq/ef,aq1n/efg,qn;q)k(aq2/efg,qn/w,qn/w,cdqn/a,a2q2/cefg,a2q2/defg,q1n/e,q1n/f,q1n/g,q;q)kqk=k=0(a2q2/cdefg,aq/fg,aq/eg,aq/ef;q)k(aq2/efg,a2q2/cefg,a2q2/defg,q;q)kqk
よってこれらを合わせると,
8ϕ7[a,aq,aq,c,d,e,f,ga,a,aq/c,aq/d,aq/e,aq/f,aq/g;a2q2cdefg]=(aq,aq/ef,aq/eg,aq/fg;q)(aq/e,aq/f,aq/g,aq/efg;q)k=0(aq/cd,e,f,g;q)k(aq/c,aq/d,efg/a,q;q)kqk+(aq,aq/ef,aq/eg,aq/fg;q)(aq/e,aq/f,aq/g,aq/efg;q)(aq/efg,aq/cd,a2q2/cefg,a2q2/defg,e,f,g;q)(a2q2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/aq;q)k=0(a2q2/cdefg,aq/fg,aq/eg,aq/ef;q)k(aq2/efg,a2q2/cefg,a2q2/defg,q;q)kqk=(aq,aq/ef,aq/eg,aq/fg;q)(aq/e,aq/f,aq/g,aq/efg;q)k=0(aq/cd,e,f,g;q)k(aq/c,aq/d,efg/a,q;q)kqk+(aq,aq/cd,a2q2/cefg,a2q2/defg,e,f,g;q)(aq/c,aq/d,aq/e,aq/f,aq/g,a2q2/cdefg,efg/aq;q)k=0(a2q2/cdefg,aq/fg,aq/eg,aq/ef;q)k(aq2/efg,a2q2/cefg,a2q2/defg,q;q)kqk
となる. 文字を置き換えることによって定理を得る.

定理1は,
8ϕ7[a,aq,aq,b,c,d,e,fa,a,aq/b,aq/c,aq/d,aq/e,aq/f;a2q2bcdef]=(aq,aq/de,aq/df,aq/ef;q)(aq/d,aq/e,aq/f,aq/def;q)4ϕ3[aq/bc,d,e,faq/b,aq/c,def/a;q]+(aq,aq/bc,d,e,f,a2q2/bdef,a2q2/cdef;q)(aq/b,aq/c,aq/d,aq/e,aq/f,a2q2/bcdef,def/aq;q)4ϕ3[aq/de,aq/df,aq/ef,a2q2/bcdefa2q2/bdef,a2q2/cdef,aq2/def;q]=(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)(aq/b,aq/c,aq/d,aq/e,aq/f,aq/def,def/a,q;q)0n(aqn+1/b,aqn+1/c,defqn/a,qn+1;q)(aqn+1/bc,dqn,eqn,fqn;q)qn+(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)(aq/b,aq/c,aq/d,aq/e,aq/f,def/aq,aq2/def;q)0n(a2qn+2/bdef,a2qn+2/cdef,aqn+2/def,qn+1;q)(aqn+1/de,aqn+1/df,aqn+1/ef,a2qn+2/bcdef;q)qn=aqdefaqdef(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)(aq/b,aq/c,aq/d,aq/e,aq/f,aq/def,def/aq,q;q)(def0n(aqn+1/b,aqn+1/c,defqn/a,qn+1;q)(aqn+1/bc,dqn,eqn,fqn;q)qnaq0n(a2qn+2/bdef,a2qn+2/cdef,aqn+2/def,qn+1;q)(aqn+1/de,aqn+1/df,aqn+1/ef,a2qn+2/bcdef;q)qn)=1defaq(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)(aq/b,aq/c,aq/d,aq/e,aq/f,aq2/def,def/a,q;q)aqdef(atq/bdef,atq/cdef,tq/def,t/a;q)(t/de,t/df,t/ef,atq/bcdef;q)dqt

と書くことができる. ここで, q積分は
abf(t)dqt:=0n(bqnf(bqn)aqnf(aqn))
によって定義されるとする.

投稿日:2024526
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