Results & Lemmas (13)
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Proposition 1.
Proposition 1. (see [6, Theorems 3]) Let f(a,b,c,d,e, x,y) be a seven-variable an- alytic function in a neighborhood of (a,b,c,d,e, x,y) =…
Proposition 1. (see [6, Theorems 3]) Let f(a,b,c,d,e, x,y) be a seven-variable an- alytic function in a neighborhood of (a,b,c,d,e, x,y) = (0,0,0,0,0,0,0) ∈C7. (I) If f(a,b,c,d,e, x,y) satisfies the following difference equation: x f(a,b,c,d,e, x,y)−f(a,b,c,d,e, x,yq) −(d +e)q−1 [ f(a,b,c,d,e, x,yq)−f(a,b,c,d,e, x,yq2)] +deq−2 [ f(a,b,c,d,e, x,yq2)−f(a,b,c,d,e, x,yq3)] = y[ f(a,b,c,d,e, x,y)−f(a,b,c,d,e, xq,y)] −(a+b+c)[ f(a,b,c,d,e, x,yq)−f(a,b,c,d,e, xq,yq)] +(ab+ac+bc)[ f(a,b,c,d,e, x,yq2)−f(
Lemma 1.
Lemma 1. Each of the following q-identities holds true: Dk a ( 1 (as;q)∞ ) = sk (as;q)∞, (1.25)
Lemma 1. Each of the following q-identities holds true: Dk a ( 1 (as;q)∞ ) = sk (as;q)∞ , (1.25)
Theorem 1.
Theorem 1. Each of the following assertions holds true: T(r, f,g,v,w,uDa) ( (as;q)∞ (az,at;q)∞ ) = (as;q)∞ (az,at;q)∞ ∞ X k=0 r, f,g, s…
Theorem 1. Each of the following assertions holds true: T(r, f,g,v,w,uDa) ( (as;q)∞ (az,at;q)∞ ) = (as;q)∞ (az,at;q)∞ ∞ X k=0 r, f,g, s z,at;q
Corollary 1.
Corollary 1. It is asserted that T(r, f,g,v,w,uDs) ( 1 (xs;q)∞ ) = 1 (xs;q)∞ 3Φ2 r, f,g; v,w; q; xu
Corollary 1. It is asserted that T(r, f,g,v,w,uDs) ( 1 (xs;q)∞ ) = 1 (xs;q)∞ 3Φ2 r, f,g; v,w; q; xu
Theorem 2.
Theorem 2. Each of the following assertions holds true: ∞ X n=0 (a;q)n a−n (q;q)n ∞ X k=0 (q−n,ax;q)k qk (q;q)k X j,i≧0 (r, f,g;q)j+i uj+i…
Theorem 2. Each of the following assertions holds true: ∞ X n=0 (a;q)n a−n (q;q)n ∞ X k=0 (q−n,ax;q)k qk (q;q)k X j,i≧0 (r, f,g;q)j+i uj+i (q;q)i (v,w;q)j+i
Theorem 3.
Theorem 3. Each of the following assertions holds true: T(r, f,g,v,w,uDx) pn x, y a;q n (cx;q)∞ (ax,bx;q)∞ = (y;q)n…
Theorem 3. Each of the following assertions holds true: T(r, f,g,v,w,uDx) pn x, y a;q n (cx;q)∞ (ax,bx;q)∞ = (y;q)n an (cx;q)∞ (ax,bx;q)∞
Corollary 2.
Corollary 2. Each of the following assertions holds true: T(r, f,g,v,w,uDx) ( xn(cx;q)∞ (ax,bx;q)∞ ) = 1 an (cx;q)∞ (ax,bx;q)∞ ∞ X k=0…
Corollary 2. Each of the following assertions holds true: T(r, f,g,v,w,uDx) ( xn(cx;q)∞ (ax,bx;q)∞ ) = 1 an (cx;q)∞ (ax,bx;q)∞ ∞ X k=0 (q−n,ax;q)k qk (q;q)k ·
Theorem 4.
Theorem 4. The following assertion holds true for y, 0: n X k=0 (q−n, x;q)k qk (q,y;q)k 3Φ2 r, f,g; v,w; q;uqk = xn y…
Theorem 4. The following assertion holds true for y , 0: n X k=0 (q−n, x;q)k qk (q,y;q)k 3Φ2 r, f,g; v,w; q;uqk = xn y x;q
Theorem 5.
Theorem 5. For m ∈N0 and y, 0, it is asserted that 2Φ1 q−n, x; y; q;q1+m = xn y x;q n (y;q)n m X j=0
Theorem 5. For m ∈N0 and y , 0, it is asserted that 2Φ1 q−n, x; y; q;q1+m = xn y x;q n (y;q)n m X j=0
Proposition 2.
Proposition 2. (see [2, Eq. (2.1)]). For max |ac|,|ad|,|bc|,|bd| < 1, it is asserted that Z d c qt c, qt d;q ∞ (at,bt;q)∞ dqt = d(1−q) …
Proposition 2. (see [2, Eq. (2.1)]). For max{|ac|,|ad|,|bc|,|bd|} < 1, it is asserted that Z d c qt c , qt d ;q ∞ (at,bt;q)∞ dqt = d(1−q) q, dq c , c
Proposition 3.
Proposition 3. (see [4, Theorems 14 and 15]) For N ∈N and r = q−N, suppose that max |ac|,|ad|,|bc|,|bd|,
Proposition 3. (see [4, Theorems 14 and 15]) For N ∈N and r = q−N, suppose that max |ac|,|ad|,|bc|,|bd|,
Theorem 6.
Theorem 6. For M ∈N and r = q−M, suppose that max |ac|,|ad|,|bc|,|bd|,
Theorem 6. For M ∈N and r = q−M, suppose that max |ac|,|ad|,|bc|,|bd|,
Theorem 7.
Theorem 7. For M ∈N and r = q−M, suppose that max |ac|,|ad|,|bc|,|bd| < 1. Then Z d c qt c, qt d;q ∞ (at,bt;q)∞ ∞ X k=0 r, f,g, c
Theorem 7. For M ∈N and r = q−M, suppose that max{|ac|,|ad|,|bc|,|bd|} < 1. Then Z d c qt c , qt d ;q ∞ (at,bt;q)∞ ∞ X k=0 r, f,g, c
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