Watsonによる8ϕ7のq-Whippleの変換公式 8ϕ7[a,aq,−aq,b,c,d,e,q−na,−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,q−naq/b,aq/c,deq−n/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,q−na,−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,q−nw,−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=efgq−n−1/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,fgq−n/a,egq−n/a;q)n(aq/e,aq/f,gq−n/a,efgq−n/a)n10ϕ9[w,wq,−wq,wb/a,wc/a,wd/a,e,f,g,q−nw,−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)∞limn→∞10ϕ9[w,wq,−wq,wb/a,wc/a,wd/a,e,f,g,q−nw,−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,=limn→∞10ϕ9[w,wq,−wq,wb/a,wc/a,wd/a,e,f,g,q−nw,−w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1;q]=limn→∞∑k=0m(w,wq,−wq,wb/a,wc/a,wd/a,e,f,g,q−n;q)k(w,−w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)kqk+limn→∞∑k=0m(w,wq,−wq,wb/a,wc/a,wd/a,e,f,g,q−n;q)n−k(w,−w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)n−kqn−k最初の項は,limn→∞∑k=0m(w,wq,−wq,wb/a,wc/a,wd/a,e,f,g,q−n;q)k(w,−w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)kqk=limn→∞∑k=0m(efgq−n−1/a,wq,−wq,aq/cd,cefgq−n−1/a2,defgq−n−1/a2,e,f,g,q−n;q)k(w,−w,cdefgq−n−1/a2,aq/c,aq/d,fgq−n/a,egq−n/a,efq−n/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,limn→∞∑k=0m(w,wq,−wq,wb/a,wc/a,wd/a,e,f,g,q−n;q)n−k(w,−w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)n−kqn−klimn→∞(w,wq,−wq,wb/a,wc/a,wd/a,e,f,g,q−n;q)n(w,−w,aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,wqn+1,q;q)nqn⋅∑k=0m(q1−n/w,−q1−n/w,bq−n/a,cq−n/a,dq−n/a,eq−n/w,fq−n/w,gq−n/w,q−2n/w,q−n;q)k(q1−n/w,q−n/w,−q−n/w,aq1−n/wb,aq1−n/wc,aq1−n/wd,q1−n/e,q1−n/f,q1−n/g,q;q)kqk係数の部分は,limn→∞(efgq−n−1/a,wq,−wq,aq/cd,cefgq−n−1/a2,defgq−n−1/a2,e,f,g,q−n;q)n(w,−w,cdefgq−n−1/a2,aq/c,aq/d,fgq−n/a,egq−n/a,efq−n/a,efg/a,q;q)nqn=limn→∞1−efgqn−1/aqn−efg/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,limn→∞∑k=0m(q1−n/w,−q1−n/w,bq−n/a,cq−n/a,dq−n/a,eq−n/w,fq−n/w,gq−n/w,q−2n/w,q−n;q)k(q1−n/w,q−n/w,−q−n/w,aq1−n/wb,aq1−n/wc,aq1−n/wd,q1−n/e,q1−n/f,q1−n/g,q;q)kqk=limn→∞∑k=0m(q1−n/w,−q1−n/w,a2q2/cdefg,cq−n/a,dq−n/a,aq/fg,aq/eg,aq/ef,aq1−n/efg,q−n;q)k(aq2/efg,q−n/w,−q−n/w,cdq−n/a,a2q2/cefg,a2q2/defg,q1−n/e,q1−n/f,q1−n/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)∞∑0≤n(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)∞∑0≤n(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=aq−defaqdef(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)∞⋅(def∑0≤n(aqn+1/b,aqn+1/c,defqn/a,qn+1;q)∞(aqn+1/bc,dqn,eqn,fqn;q)∞qn−aq∑0≤n(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)=1def−aq(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:=∑0≤n(bqnf(bqn)−aqnf(aqn))によって定義されるとする.
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