On a new parameter involving Ramanujan's theta-functions
aa r X i v : . [ m a t h . N T ] A ug ON A NEW PARAMETER INVOLVING RAMANUJAN’S THETA-FUNCTIONS
S. CHANDANKUMAR AND H. S. SUMANTH BHARADWAJA
BSTRACT . We define a new parameter A ′ k,n involving Ramanujan’s theta-functions forany positive real numbers k and n which is analogous to the parameter A k,n defined byNipen Saikia [8]. We establish some modular relation involving A ′ k,n and A k,n to findsome explicit values of A ′ k,n . We use these parameters to establish few general theoremsfor explicit evaluations of ratios of theta functions involving ϕ ( q ) .
1. I
NTRODUCTION
Let ( a ; q ) ∞ denote the infinite product ∞ Y n =0 (1 − aq n ) ( a , q are complex numbers, | q | < ). Following theta-functions ϕ , ψ and f with | q | < are classical: ϕ ( q ) := f ( q, q ) = ∞ X n = −∞ q n = ( − q ; q ) ∞ ( q ; q ) ∞ , (1.1) ψ ( q ) := f ( q, q ) = ∞ X n =0 q n ( n +1) / = ( q ; q ) ∞ ( q ; q ) ∞ , (1.2) f ( − q ) := f ( − q, − q ) = ∞ X n = −∞ ( − n q n (3 n − / = ( q ; q ) ∞ . (1.3)which follows from the Ramanujan’s general theta-function defined by f ( a, b ) := ∞ X n = −∞ a n ( n +1) / b n ( n − / , | ab | < , using Jacobi’s fundamental factorization formula f ( a, b ) can be expressed in product as f ( a, b ) = ( − a ; ab ) ∞ ( − b ; ab ) ∞ ( ab ; ab ) ∞ . The ordinary or Gaussian hypergeometric function is defined by F ( a, b ; c ; z ) := ∞ X n =0 ( a ) n ( b ) n ( c ) n n ! z n , ≤ | z | < , Mathematics Subject Classification.
Key words and phrases.
Modular equations, theta-functions. where a , b , c are complex numbers, c = 0 , − , − , . . . , and ( a ) = 1 , ( a ) n = a ( a + 1) · · · ( a + n − for any positive integer n . Now we recall the notion of a modular equation. Let K ( k ) be the complete elliptic integralof the first kind of modulus k . Recall that K ( k ) := Z π dφ p − k sin φ = π ∞ X n =0 (cid:0) (cid:1) n ( n !) k n = π ϕ ( q ) , (0 < k < , (1.4)and set K ′ = K ( k ′ ) , where k ′ = √ − k is the so called complementary modulus of k .It is classical to set q ( k ) = e − πK ( k ′ ) /K ( k ) so that q is one-to-one and increases from 0 to1. Setting α = k one knows that ϕ ( − q ) = ϕ ( q )(1 − α ) / , (1.5) ϕ ( − q ) = ϕ ( q )(1 − α ) / . (1.6)These formulas correspond to those recorded by Ramanujan in Chapter 17 of his secondnotebook [5, Ch. 17, Entry 10 (ii), (iii), p. 122]. In the same manner introduce L = K ( ℓ ) , L ′ = K ( ℓ ′ ) and suppose that the following equality n K ′ K = L ′ L , (1.7)holds for some positive integer n . Then a modular equation of degree n is a relationbetween the moduli k and ℓ which is induced by (1.7). Following Ramanujan, set α = k and β = ℓ . We say that β is of degree n over α . The multiplier m , corresponding to thedegree n , is defined by m := KL = ϕ ( q ) ϕ ( q n ) , (1.8)for q = e − πK ( k ′ ) /K ( k ) . By using the transformation formulae for theta-functions recordedby Ramanujan for theta-functions f ( q ) and ϕ ( q ) , J. Yi [10] introduced parameters r k,n and r ′ k,n as follows: r k,n = f ( − q ) k / q ( k − / f ( − q k ) , where q = e − π √ n/k , (1.9) r ′ k,n = f ( q ) k / q ( k − / f ( q k ) , where q = e − π √ n/k . (1.10)She also introduced h k,n and h ′ k,n as follows: h k,n := ϕ ( q ) k / ϕ ( q k ) , where q = e − π √ n/k , (1.11) N A NEW PARAMETER INVOLVING RAMANUJAN’S THETA-FUNCTIONS 3 h ′ k,n := ϕ ( − q ) k / ϕ ( − q k ) , where q = e − π √ n/k , (1.12)and systematically studied several properties and also found plethora of explicit evaluationsof r k,n , r ′ k,n , h k,n and h ′ k,n for different positive real values of n and k .In [1], the authors derived a new transformation formula for theta-function ψ ( − q ) , thistransformation formula was studied by Baruah and Nipen Saikia [3] and Yi, Y. Lee and D.H. Paek [12] through parameters l k,n and l ′ k,n as follows: l k,n := ψ ( − q ) k / q ( k − / ψ ( − q k ) , where q = e − π √ n/k (1.13)and l ′ k,n := ψ ( q ) k / q ( k − / ψ ( q k ) , where q = e − π √ n/k . (1.14)and established several evaluations of l k,n and l ′ k,n different positive real values of n and k . Recently, Nipen Sikia [8] by using one of the transformation formula recorded by Ra-manujan introduced the following parameter A k,n as A k,n = ϕ ( − q )2 k / q k/ ψ ( q k ) , q = e − π √ n/k . (1.15)where n and k are positive real numbers. He studied several properties and establishedsome general theorems for the explicit evaluations of A k,n .We define a new parameter A ′ k,n for any positive real numbers n and k as: Definition 1.1.
For and positive rationals n and k , define A ′ k,n = ϕ ( q )2 k / q k/ ψ ( q k ) , q = e − π √ n/k . (1.16)This work is organized as follows. Some notations and background results are listed inSection 2. We derive new modular equations involving theta-functions ϕ and ψ in Section3. Several new explicit evaluations of A k,n and A ′ k,n , for some positive rationals n and k are established in section 4. In Section 5 we establish some new modular relations fora modular function of Level 16 developed by Dongxi Ye and also explicitly evaluate thefunction. Finally, in Section 6 we list some general formulas for explicit evaluations of h ′ ,n and l ′ ,n and give few examples. S. CHANDANKUMAR AND H. S. SUMANTH BHARADWAJ
2. P
RELIMINARY RESULTS
In this section, we list few theta-function identities involving theta-functions ϕ and ψ which are useful in deriving modular equations. Lemma 2.1.
We have [5, Entry 25 (ii), (iii), (iv), (v) and (vii), p. 40] ϕ ( q ) − ϕ ( − q ) = 4 qψ ( q ) , (2.1) ϕ ( q ) ϕ ( − q ) = ϕ ( − q ) , (2.2) ϕ ( q ) ψ ( q ) = ψ ( q ) , (2.3) ϕ ( q ) − ϕ ( − q ) = 8 qψ ( q ) , (2.4) ϕ ( q ) − ϕ ( − q ) = 16 qψ ( q ) . (2.5) Lemma 2.2. [5, Entry 11 (i), (iii), (iv) and (v), pp. 122]
We have ψ ( q ) = r z (cid:18) αq (cid:19) / , (2.6) ψ ( q ) = √ z (cid:18) αq (cid:19) / , (2.7) ψ ( q ) = √ z { − √ − α } / √ q , (2.8) ψ ( q ) = √ z { − (1 − α ) / } q , (2.9)where z := F (cid:0) , ; 1; x (cid:1) and y := π F (cid:0) , ; 1; 1 − x (cid:1) F (cid:0) , ; 1; x (cid:1) . Lemma 2.3.
We have(i) [5, Eq. (24.21), p. 215] If β is of degree over α , then β = (cid:18) − √ − α √ − α (cid:19) . (2.10) (ii) [5, Entry 5 (ii), p. 230] If β is of degree over α , then ( αβ ) / + ((1 − α )(1 − β )) / = 1 . (2.11) (iii) [5, Entry 13(i), p. 280] If β is of degree over α , then ( αβ ) / + ((1 − α )(1 − β )) / + 2 ( αβ (1 − α )(1 − β )) / = 1 . (2.12) N A NEW PARAMETER INVOLVING RAMANUJAN’S THETA-FUNCTIONS 5 (iv) [5, Entry 19, p. 314] If β is of degree over α , then ( αβ ) / + ( αβ (1 − α )(1 − β )) / = 1 . (2.13) Lemma 2.4. [5, p. 233, (5.2)] If m = z /z and β has degree 3 over α , then β = ( m − (3 + m )16 m (2.14) Lemma 2.5. [5, Entry 10 (i), pp. 122]
We have ϕ ( q ) = √ z. (2.15)We end this section by listing few values of r ,n found out by Yi in her thesis [10]. Lemma 2.6.
We have r , = 2 / , r , = 2 / ( √ / , r , = (1 + √ / , r , = ( √ √ / , r , = 2 / ( √ / , r , = 2 / ( √ / , r , = r √ , r , = s √ p √ .
3. S
OME NEW N EW MODULAR EQUATIONS
In this section, we establish few theta-function identities involving ϕ and ψ which playkey role in establishing some general Theorems for explicit evaluations is A ,n , A ′ ,n , and h ′ ,n for some rationals n . Lemma 3.1. If P := ϕ ( − q ) qψ ( q ) and R := ϕ ( q ) qψ ( q ) , then R = P + 4 . (3.1) Proof.
Invoking (2.1), we arrive at (3.1). (cid:3)
Lemma 3.2. If P := ϕ ( − q ) √ qψ ( q ) and R := ϕ ( q ) √ qψ ( q ) , then R = P + 8 . (3.2) Proof.
Using (2.4), we arrive at (3.2). (cid:3)
Lemma 3.3. If P := ϕ ( − q ) q / ψ ( q ) and R := ϕ ( q ) q / ψ ( q ) , then R = P + 16 . (3.3) Proof.
Using (2.5), we arrive at (3.3). (cid:3)
S. CHANDANKUMAR AND H. S. SUMANTH BHARADWAJ
Lemma 3.4. If P := ϕ ( − q ) q / ψ ( q ) and R := ϕ ( q ) q / ψ ( q ) , then R = P + 16 R . (3.4) Proof.
Using (2.5), we arrive at (3.3). (cid:3)
Lemma 3.5. If P := ϕ ( − q ) ϕ ( − q ) and Q := ϕ ( − q ) qψ ( q ) , then ( Q + 4) P = Q. (3.5) Proof.
Equation (2.2) can be rewritten as ϕ ( − q ) ϕ ( − q ) = ϕ ( − q ) ϕ ( q ) . (3.6)Using (2.1) in the above equation, we arrive at (3.6). (cid:3) Lemma 3.6. If P := ϕ ( − q ) ϕ ( − q ) and Q := ϕ ( − q ) √ qψ ( q ) , then ( Q + 8) P = Q . (3.7) Proof.
Squaring equation (3.6), we have ϕ ( q ) ϕ ( − q ) = ϕ ( − q ) . (3.8)Using (2.4) in the above equation, we arrive at (3.7). (cid:3) Lemma 3.7. If P := ϕ ( − q ) ϕ ( − q ) and Q := ϕ ( − q ) √ qψ ( q ) , then ( Q + 16) P = Q . (3.9) Proof.
Squaring equation (3.8), we have ϕ ( q ) ϕ ( − q ) = ϕ ( − q ) . (3.10)Using (2.5) in the above equation, we arrive at (3.10). (cid:3) Lemma 3.8. If P := ϕ ( − q ) ϕ ( − q ) and Q := ϕ ( − q ) √ qψ ( q ) , then ( Q + 16 P ) P = Q . (3.11) N A NEW PARAMETER INVOLVING RAMANUJAN’S THETA-FUNCTIONS 7
Proof.
From (1.5) and (1.6), we have α = 1 − P . (3.12)Using equations (1.5) and (2.6), we get Q = 16 (1 − α ) α (3.13)Substituting for α from (3.12) in above equation, we arrive at (3.11). (cid:3) Theorem 3.1. If P := ϕ ( q ) q / ψ ( q ) and Q := ϕ ( q ) ϕ ( q ) , then ( Q − Q + 8 Q − P = 256 Q . (3.14) Proof.
Using (2.7) and (2.15), we have P = 2 √ mβ / and Q = √ m. (3.15)Invoking (2.14) the above equation can be written as P ( Q − (3 + Q )16 Q ) = (2 Q ) . (3.16)On factorizing above equation, we arrive at (3.14) which completes the proof. (cid:3) Remark . By transcribing (3.14) using the definition of A ′ ,n and h ,n one can findevaluation of A ′ ,n for some positive rationals n by using the values of h ,n . Theorem 3.3. If P := ϕ ( − q ) qψ ( q ) and Q := ϕ ( − q ) q ψ ( q ) , then P + 8 Q + 2 PQ + 4 = QP . (3.17)
Proof.
Transcribing P and Q by using (1.5) and (2.9), we obtain β = 1 − (cid:18) QQ + 4 (cid:19) and √ − α = (cid:18) PP + 4 (cid:19) . (3.18)Employing (3.18) in the equation (2.10), we arrive at ( − Q + 4 QP + 8 P + QP + 2 P )( Q P + 4 Q P + 4 Q + 16 QP + 32 P + 4 QP + 8 P ) . (3.19)It is observed that for | q | < , the second factor Q P + 4 Q P + 4 Q + 16 QP + 32 P +4 QP + 8 P = 0 . Thus the first factor i.e., QP + 8 P + QP + 2 P − Q = 0 . S. CHANDANKUMAR AND H. S. SUMANTH BHARADWAJ
Dividing the above equation by
P Q and then rearranging, we arrive at (3.17) to completethe proof. (cid:3)
Throughout this section, we set A n := ϕ ( − q ) ϕ ( − q n ) q n +1 ψ ( q ) ψ ( q n ) and B n := ϕ ( − q ) ψ ( q n ) q − n ϕ ( − q n ) ψ ( q ) (3.20) Theorem 3.4.
We have B + 1 B = 12 (cid:18) B + 1 B (cid:19) + (cid:18) A + 8 A (cid:19) + 6 (cid:18)p A + 8 √ A (cid:19) (cid:18)p B + 1 √ B (cid:19) + 30 . (3.21) Proof.
Let us begin the proof by setting P := ϕ ( − q ) qψ ( q ) and Q := ϕ ( − q ) q ψ ( q ) . Transcribing P and Q by using the (1.5) and (2.9), we obtain β = 1 − (cid:18) QQ + 4 (cid:19) and α = 1 − (cid:18) PP + 4 (cid:19) . (3.22)Ramanujan’s modular equations of degree three in (2.11) can be written as αβ = (cid:16) − { (1 − α )(1 − β ) } / (cid:17) . (3.23)Again invoking (1.5) and (2.9) in the above equality and set A := P Q and B = P/Q ,we arrive at (3.21) to complete the proof. (cid:3)
Theorem 3.5.
We have B + 1 B − (cid:18) B + 1 B (cid:19) + 785 (cid:18) B + 1 B (cid:19) + (cid:18) A + 8 A (cid:19) + 80 (cid:18)p A + 8 A (cid:19) " (cid:18)p B + 1 √ B (cid:19) + q B + 1 p B ! + 20 (cid:18) A + 8 A (cid:19) (cid:20) (cid:18) B + 1 B (cid:19)(cid:21) + 10 q A + 8 p A ! (cid:18)p B + 1 B (cid:19) . (3.24) Proof.
The proof of the (3.24) is similar to the proof of the equation (3.22), except that weuse Ramanujan’s modular equations of degree five (2.12), hence we omit the proof. (cid:3)
N A NEW PARAMETER INVOLVING RAMANUJAN’S THETA-FUNCTIONS 9
Theorem 3.6.
We have B + 1 B = 280 (cid:18) B + 1 B (cid:19) + 9772 (cid:18) B + 1 B (cid:19) + 60424 (cid:18) B + 1 B (cid:19)(cid:18) A + 8 A (cid:19) + (cid:18) A + 8 A (cid:19) (cid:20)
203 + 84 (cid:18) B + 1 B (cid:19)(cid:21) + 28 (cid:18)p A + 8 √ A (cid:19) × " (cid:18) √ B + 1 √ B (cid:19) + 313 q B + 1 p B ! + 21 q B + 1 p B ! + 140 q A + 8 p A ! " (cid:18) √ B + 1 √ B (cid:19) + 2 q B + 1 p B ! + (cid:18) A + 8 A (cid:19) (cid:20) (cid:18) B + 1 B (cid:19) + 546 (cid:18) B + 1 B (cid:19)(cid:21) + 14 (cid:18)q A + 8 A (cid:19) (cid:18)p B + 1 B (cid:19) + 106330 (3.25) Proof.
The proof of the (3.24) is similar to the proof of the equation (3.22), except that weuse Ramanujan’s modular equations of degree seven (2.13), hence we omit the proof. (cid:3)
4. E
XPLICIT EVALUATION OF A k,n AND A ′ k,n In this section, we establish some general theorems for explicit evaluation of A k,n and A ′ k,n by using the modular equations established in Section 3. Lemma 4.1.
For any positive rational n , we have A / ,n = r ,n (4.1) Proof.
By Entry 24 (iii) [5, p. 39], we have ϕ ( q ) ψ ( q ) = f ( − q ) f ( − q ) . (4.2)By using the definition of A k,n for k = r k,n for k =
2, we complete the proof of(4.1). (cid:3)
Lemma 4.2.
Let n and k be any two positive rational such that k > , we have A ′ k/ ,n = r , n r k, n r ,k n r ,n . (4.3) Proof.
By using the definition of theta functions ϕ ( q ) , ψ ( q ) and f ( − q ) , we have ϕ ( q ) ψ ( q k ) = ( q ; q ) ∞ ( q k ; q k ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q k ; q k ) ∞ = f ( − q ) f ( − q ) f ( − q ) f ( − q ) f ( − q ) f ( − q k ) f ( − q k ) f ( − q k ) . (4.4)By using the definitions of A ′ k,n and r k,n in (4.4), we arrive at required result. (cid:3) Theorem 4.1.
We have A / , = 2 / , (4.5) A / , = q √ , (4.6) A / , = q √ , (4.7) A / , = vuut
12 + √ ! , (4.8) A / , = 2 / q ( √ , (4.9) A / , = vuut √
22 + 12 + p √ − ! , (4.10) A / , = 2 / q ( √ , (4.11) A / , = √ √ . (4.12) Proof.
The proof of the above Theorem follows from Lemma 4.1 and the correspondingvalues of r ,n from Lemma 2.6. (cid:3) Theorem 4.2.
For any positive real number n , we have(i) A ′ ,n = A ,n + √ , (ii) A ′ ,n = q A ,n + √ , (iii) A ′ ,n = q A ,n + 1 , (iv) A ′ / ,n ) − ( A ′ / ,n ) − = 4 A / ,n . Proof.
The above Theorem can be proved by transcribing the modular equations estab-lished in Section 3 along with the definition of A ′ k,n and A k,n .To prove Theorem 4.2 (i), we use the definition of A k,n and A k,n with n = 4 and(3.1). Similarly Theorem 4.2 (ii) follows from (3.2), Theorem 4.2 (iii) follows from (3.3),Theorem 4.2 (iv) follows from (3.4). (cid:3) N A NEW PARAMETER INVOLVING RAMANUJAN’S THETA-FUNCTIONS 11
Theorem 4.3.
We have A , = 1 + q √ , (4.13) A , = (1 + √ √ √ , (4.14) A , = q √ q √ , (4.15) A , = (3 + √ √ , (4.16) A , = q
52 + 36 √ √ a + q
56 + 40 √ √ a, (4.17) A , = q
94 + 66 √ √ √ q
93 + 66 √ √ √ , (4.18) A , = q
402 + 232 √ √ √ q
416 + 240 √ √ √ , (4.19) A , = q √
14 + 24432 √ √ q (32384 + 22899 √ √ √ , (4.20) where a = p √ . To establish the values of A ,n listed in the above Theorem, we use the definition of A k,n for k = Proof of (4.13) . Transcribing (3.17) by using the definition of A ,n , we have A n = 4 A n A n + 2 √ A n + 2 √ A n A n + 2 A n , (4.21)where A n = A ,n and A n = A , n .Set n = 1 / in the above equaton and using the fact that A , A , / = 1 , we have ( h − h − √ h + 2 h + √
2) = 0 , where h := A , . (4.22)Since the roots of the second factor has no real roots, hence on solving h − h − √ and since A , > , we arrive at (4.13) to complete the proof. (cid:3) Proof of (4.14) . Transcribing (3.21) by using the definition of A ,n , we have A n A n + 12 √ A n A n + 12 A n A n + 12 √ A n A n + 30 A n A n + 12 √ A n A n + 12 A n A n + 12 √ A n h + 8 hA n = h + A n , (4.23) where A n = A ,n and A n = A , n .Set n = 1 / in the above equation and using the fact that A , A , / = 1 , we have ( h − h − √ h − − √ h + 2 h − √ h − √ h + √ h + 1) = 0 , (4.24)where h := A , . Since A , > , hence on solving h − (2 + √ h − − √ and since A , > ,we arrive at (4.14) to complete the proof. (cid:3) The values of A , /n for n =
2, 3, 4, 7, 8, 9, 12 and 28 can easily be found by using thefact that A ,n A , /n = 1 . Proof of (4.15) . Setting n = 1 in (4.21) and using the fact that A , = 1 , we get h = 4 h + 2 √ √ h + 2 , where h := A , . (4.25)Since A , > , hence on solving the above equation, we arrive at (4.15) to completethe proof. (cid:3) Proof of (4.16) . Transcribing A and B , defined in (3.20) along with the definition of A ,n , we have A = 8 hH and B = h/H, (4.26)where H = A ,n and h = A , n .Set n = 1 / in the above equation and knowing the fact that A , A , / = 1 , then A = 8 and B = 1 /H . Using A and B in (3.25) and factorizing, we arrive at ( H − H − √ H + 1)( H + 12 H − √ H + 1)( H + √ H + 1) ( H + 2 H + 3 √ H + 1) ( H − H + 3 √ H + 1) = 0 . (4.27)where H = A , .Since A , > , hence on solving H − (12 + 7 √ H + 1 = 0 , we arrive at (4.16) tocomplete the proof. (cid:3) We observe that (4.21) results in a quadratic equation in A , n for any know value of A ,n . We use the value of A , to get A , , A , to get A , , A , to get A , respectively.Hence we omit the proof. The values of A , /n where n ∈ { , , , , , , , } can beeasily found out by the fact that A ,n A , /n = 1 . N A NEW PARAMETER INVOLVING RAMANUJAN’S THETA-FUNCTIONS 13
Theorem 4.4.
We have A ′ , = 1 + √ , (4.28) A ′ , = 1 + √ q √ , (4.29) A ′ , = ( √ √ √ √ , (4.30) A ′ , = q
12 + 8 √ q √ (4.31) A ′ , = (3 + √ √ √ (4.32) A ′ , = q
108 + 62 √ √ √ q
93 + 66 √ √ √ , (4.33) A ′ , = q
402 + 232 √ √ √ q
416 + 240 √ +294 √ √ (4.34) A ′ , = 2 √ (cid:18)q
62 + 44 √ √ a + q
56 + 40 √ √ a (cid:19) , (4.35) A ′ , = √ √
14 + 24432 √ √ + 2(32384 + 22899 √ √ √ , (4.36) where a = p √ . Proof.
For proof of Theorem 4.4, we use Theorem 4.2(i) and corresponding values of A ,n in Theorem 4.3 to complete the proof. (cid:3)
5. M
ODULAR FUNCTION OF LEVEL | q | < ,h ( q ) = q ∞ Y j =1 (1 − q j ) (1 − q j )(1 − q j ) (1 − q j ) . (5.1)In his work, he established some basic properties involving h and Ramanujan’s theta func-tion. In the following Lemma, we list one of the relation Lemma 5.1.
We have h ( q ) = qψ ( q ) ϕ ( − q ) . (5.2) He established modular relation for h ( q ) connecting with h ( − q ) , h ( q ) , h ( q ) and h ( q ) . Note that in Section 3, we have provided algebraic relations between ϕ ( − q ) qψ ( q ) and ϕ ( − q n ) q n ψ ( q n ) for n ∈ and using which one can establish relation connecting h ( q ) with h ( q n ) , for n ∈ and . We list the relation in the following Theorem, set u := h ( q ) and v n := h ( q n ) then Theorem 5.1.
We havei) u + ( − v − v − v ) u + ( − v − v − v ) u + ( − v − v − v ) u + v = 0 . ii) (10 u + 80 u + 70 u + u + 40 u ) v + (4096 u + 640 u + 70 u + 5120 u +2560 u ) v +(6400 u +785 u +80 u +3200 u +5120 u ) v +(3200 u +400 u +1620 u +40 u +2560 u ) v +(100 u +10 u +785 u +640 u +400 u ) v = v + u . Proof.
Theorem 5.1 follows from the definition of h ( q ) and equations (3.21) and (3.24). (cid:3) Theorem 5.2.
For any positive real number n , we have h ( e − π √ n ) = 1 √ A ,n . (5.3) Proof.
By using the definition of h ( q ) and A ,n , we arrive at (5.3). (cid:3) Lemma 5.2.
We have h ( e − π √ ) = 1 √ , (5.4) h ( e − π ) = p √ − , (5.5) h ( e − π ) = p √ √ , (5.6) h ( e − π √ ) = ( √ − √ − , (5.7) h ( e − π √ ) = ( √ √ , (5.8) h ( e − π √ ) = q √ − p √ √ √ , (5.9) h ( e − π √ ) = q √
2) + p √ √ . (5.10) N A NEW PARAMETER INVOLVING RAMANUJAN’S THETA-FUNCTIONS 15
The evaluations listed in above Lemma follows easily by using the values of A ,n for n =
1, 2, 1/2, 3, 1/3, 4 and 1/4 respectively established in Theorem 4.3 in (5.3).6. E
XPLICIT EVALUATION OF h ′ ,n In this section, we list applications of modular equations found in Section 3 to derivefew relations connecting the parameters A k,n with h ′ ,n . We begin this section by listingfew explicit evaluations of h ′ ,n for different real number n. Theorem 6.1.
We have h ′ , / = 2 − / q √ − , (6.1) h ′ , / = 2 − / q ( √ √ √ , (6.2) h ′ , / = 2 − / ( √ − q √ √ , (6.3) h ′ , / = (cid:2) ( √ − √ (4 + (12 − √ a ) + 12 √ − √ (cid:3) / , (6.4) h ′ , / = p √ − √ , (6.5) h ′ , / = 2 − / q ( √ − √ − √ , (6.6) h ′ , / = q (8 + 6 √ − p √ , (6.7) h ′ , / = 2 − / ( √ − q √ − √ , (6.8) where a := p
93 + 66 √ √ √ . Proof.
When we transcribe (3.5) by using the definition of h ′ ,n and A ,n , we get √ A + 2 h = A, (6.9)where h := h ′ ,n/ and A := A ,n . Note that the above equation is general formula to explicitly evaluate h ,n/ for anypositive rationals n . We prove few values listed in above Theorem 6.1. For proof of (6.1),letting n = 1 in (6.9) we know that A , = 1 , we find that h , / + √ . (6.10)Solving the above equation for h , / and using the fact that h , / > , we complete theproof. For (6.2)–(6.8), we repeat the same argument as in the proof of (6.1) to complete theproof. (cid:3)
Theorem 6.2.
We have h ′ , / = 2 − / q √ − , (6.11) h ′ , / = q (16 − √ √ √ , (6.12) h ′ , / = q (16 − √ √ − √ , (6.13) h ′ , / = q ( √ − (56 + 32 √ , (6.14) h ′ , / = q ( √ − (56 − √ . (6.15) Proof.
When we transcribe (3.7) by using the definition of h ′ ,n and A ,n , we get A h + 2 √ h = A , (6.16)where h := h ′ ,n/ and A := A ,n . Note that the above equation is general formula toexplicitly evaluate h ,n/ for any positive rationals n . For brevity we prove (6.11) and(6.12). Put n = 1 in (6.16) and using the fact that A , = 1 , we have √ h + 1 , (6.17)On solving the above equation and since h ′ , / > , we arrive at (6.11).For proof of (6.12), we let n = 3 and using the value of A , = q ( √ √ √ found by Saikia [8] in (6.16), we get h + √ − √ − √ . (6.18)On solving the above equation and noting that h , / > , we arrive at the (6.12). (cid:3) For (6.13)-(6.15), we repeat the same argument as in the proof of (6.12) except we usethe values of A , / , A , and A , / respectively obtained by Saikia [8]. Theorem 6.3. If h := h ′ ,n/ and A := A ,n then A + 1) h = A . (6.19) Proof.
Transcribing (3.9) by using the definition of h ′ ,n and A ,n , we arrive at (6.19). (cid:3) N A NEW PARAMETER INVOLVING RAMANUJAN’S THETA-FUNCTIONS 17
Theorem 6.4. h ′ , = 2 / ( √ − / , (6.20) h ′ , = 2 − / ( − − √ √ √ / , (6.21) h ′ , = 2 − / ( − − √ √
10 + 114 √ / , (6.22) h ′ , = (85 √ √ − − √ / , (6.23) h ′ , = 2 − / (2772 √ − √ − √ / . (6.24) Proof.
When we transcribe (3.11) by using the definition of h ′ ,n and A / ,n , we get (4 h − A + 4 h = 0 , (6.25)where h := h ′ ,n and A := A / ,n Note that the above equation is general formula toexplicitly evaluate h ,n for any positive rationals n . For brevity we prove (6.20). Put n = 1 in (6.25) and using the fact that A / , = 1 , we have h + h = 14 . (6.26)On solving the above equation and since h ′ , > , we arrive at (6.20).For proving (6.21)-(6.24), we use (6.25) along with the corresponding value of A / ,n and repeat the same argument as in the proof of (6.12), to complete the proof. (cid:3) R EFERENCES[1] C. Adiga, Taekyun Kim, M. S. Mahadeva Naika and H. S. Madhusudhan, On Ramanujan’s cubic continuedfraction and explicit evaluations of theta-functions, Indian J. pure appl. math., 35(9) (2004), 1047–1062.[2] G.E.Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part I, Springer, New York, 2005.[3] N. D. Baruah and Nipen Saikia, Two parameters for Ramanujan’s theta–functions and their explicit values,Rocky Mountain J. Math., 37(6) (2007), 1747–1790.[4] B. C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan, J.Reine Angew. Math. 303/304 (1978) 332-365.[5] B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer–Verlag, New York, 1991.[6] B. C. Berndt, Ramanujan’s Notebooks, Part V, Springer–Verlag, New York, 1998.[7] S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.[8] Nipen Saikia, A new parameter for Ramanujan’s theta-functions and explicit values, Arab J. Math. Sci., 18(2) (2012), 105–119.[9] D. Ye, Level 16 analogue of Ramanujan’s theories of elliptic functions to alternative bases, J. Number Theory,164 (2016), 191–207. [10] J. Yi, Construction and application of modular equations, Ph.D thesis, University of Illinois at Urbana-Champaign, 2004.[11] J. Yi, Theta-function identities and the explicit formulas for theta-function and their applications, J. Math.Anal. Appl., 292(2004), 381-400.[12] J. Yi, Yang Lee and Dae Hyun Paek, The explicit formulas and evaluations of Ramanujan’s theta-function ψ , J. Math. Anal. Appl., 321 (2006), 157–181.(S. Chandankumar and H. S. Sumanth Bharadwaj) D EPARTMENT OF M ATHEMATICS , M. S. R
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