New theta-function identities and general theorems for the explicit evaluations of Ramanujan’s continued fractions

We prove some new theta-function identities for two continued fractions of Ramanujan which are analogous to those of Ramanujan–Göllnitz–Gordon continued fraction. Then these identities are used to prove new general theorems for the explicit evaluations of the continued fractions.

( 1.11) and , |q| < 1. (1.12) and use them to prove new general theorems for the explicit evaluations of T (q) and W (q). The continued fractions T (q) and W (q) are introduced and studied by Saikia in [14] and [15], respectively. Saikia [14, p. 4, Theorem 3.1] proved that The identity analogous to (1.13) and satisfied by the continued fraction W (q) is [15, Theorem 3.1]: (1.14) Saikia also established some modular relations and explicit values for T (q) and W (q) in [14] and [15], respectively. In Sects. 3 and 4, we prove new theta-function identities for the continued fractions T (q) and W (q), respectively. In Sects. 5 and 6, we prove new general theorems for the explicit evaluations of T (q) and W (q) by using theta-function identities established in Sects. 3 and 4, respectively, and give examples of explicit evaluations. Section 2 is devoted to record some preliminary results for ready references in this paper.
To end this introduction, we define some parameters of theta-functions which will be used in the explicit evaluations of T (q) and W (q). For any positive real numbers k and n, define The parameter A k,n is introduced by Saikia [13, p. 107, (1.7)]. The parameter s 4,n is the particular case k = 4 of the general parameter s k,n defined by and is due to Berndt [6, p. 9, (4.7)]. The parameter J n is the particular case k = 4 of the general parameter r k,n , introduced by Yi [20, p. 11, (2.1.1)] (also see [6, p. 9, (4.6)]) and defined by Yi [20] evaluated several explicit values of the parameter r k,n .

Preliminary results
This section is devoted to record some transformation formulas and P-Q theta-function identities which will be used in the succeeding sections. The P-Q identities presented in Lemmas 2.6-2.8 are new. Since modular equations are key in the proofs of P-Q theta-function identities, first we define Ramanujan's modular equation. The ordinary or Gaussian hypergeometric function 2 F 1 (a, b; c; x) is defined by where (a) 0 = 1 and (a) n = a(a + 1)(a + 2) · · · a(a + n − 1) for n ≥ 1 and |x| < 1. Let, for 0 < α < 1, Assume that for some integer n Then a modular equation of degree n is a relation between α and β induced by (2.2). We often say β has degree n over α. The multiplier m connecting α and β is defined by m = z 1 /z n , where z n = 2 F 1 1 2 , 1 2 ; 1; β .
Proof Transcribing using Lemma 2.5, we find that where β has degree 2 over α. From [5, p. 214, Entry 24(iii)] we note that, if β has degree 2 over α, then and (2.11) Eliminating m between (2.10) and (2.11) and then simplifying, we deduce that Employing (2.9) in (2.12) and factorizing using Mathematica, we obtain (2.13) Since the second factor is non-zero, we arrive at the desired result. (2.14) Proof Transcribing using Lemma 2.5, we find that and where β has degree 3 over α. From [5, p. 230, Entry 5(ii)] we note that, if β has degree 3 over α, then and can also be expressed as Employing (2.15) and (2.16) in (2.18) and simplifying with the help of Mathematica, we arrive at the desired result.
Proof Transcribing using Lemma 2.5, we find that where β has degree 5 over α. From [5, p. 280-281, Entry 13(v) & (vi)] we note that, if β has degree 5 over α, then , (2.24) respectively. Eliminating m between (2.23) and (2.24) and simplifying, we deduce that Squaring (2.26) and substituting for c and d from (2.20) and simplifying with the help of Mathematica, we arrive at desired result.

New identities for T (q)
In this section we prove theta-function identities for T (q) analogous to (1.7)-(1.10).

Theorem 3.1 We have
Proof (i) From (1.13), we note that From Lemma 2.4, we note that Employing (3.2) in (3.1), and simplifying, we deduce that Employing (2.3) and (2.4) in (3.3) and simplifying, we obtain Replacing q by q 1/4 in (3.4), we prove the first equality. To prove the second equality, from Lemma 2.4 we note that .
Employing (3.5) in the first equality, we arrive at the desired result. .
From Lemma 2.4, we note that Employing (2.4) and (3.8) in (3.7) and simplifying, we obtain Replacing q by q 1/2 in (3.9), we arrive at the desired result.

New identities for W (q)
This section is devoted to proving theta-function identities analogous to (1.7)-(1.10) for the continued fraction W (q).

Theorem 4.1 We have
Proof (i) From (1.14), we deduce that Squaring (3.2), then employing in (4.1) and simplifying, we obtain Employing (2.5) and (2.6) in (4.2) and simplifying, we obtain Replacing q by √ q in (4.3) we prove the first equality. To prove the second equality, from Lemma 2.4 we note that Employing (4.4) in the first equality, we arrive at the desired result.
(4.6) Employing (2.5) and (2.7) in (4.6) and simplifying, we obtain Replacing q by √ q in (4.7), we prove the first equality. To prove the second equality, from Lemma 2.4 we note that Employing (4.8) in the first equality, we arrive at the desired result.
(iii) From part (i) and (ii), we deduce that Employing (2.7) in (4.9) and simplifying, we complete the proof.

Corollary 4.3 We have
Proof Squaring Theorem 4.2(i) and (ii) we easily arrive at (i) and (ii), respectively.

General theorems for explicit evaluations of T (q)
In this section we prove new general theorems for the explicit evaluations of T (q) and give examples.
Theorem 5. 5 We have Proof Setting q = e −π √ n/4 in Theorem 3.1(iv) and employing the definition of A k,n with k = 4, we arrive at (i). Replacing n by 1/n in (i) and simplifying using Lemma 2.1, we complete the proof of (ii).
A systematic study of the parameter A k,n for k = 4 has not been undertaken and no other value of the parameter A 4,n is evaluated in literature. So we devote the remainder of this section to evaluate some new explicit values of A 4,n by using P-Q theta-function identities established in Sect. 2.
Theorem 5. 7 We have . Solving (5.4) using Mathematica and noting A 4,n has positive real value greater than unity, we arrive at (i).
Theorem 5. 8 We have Solving (5.9) using Mathematica and choosing the appropriate root, we complete the proof of (ii).
Theorem 5. 9 We have  4,25 , and choosing the appropriate root, we complete the proof of (ii).

General theorems for the explicit evaluations of W (q)
In this section we prove general theorems for the explicit evaluations of the continued fraction W (q).
Proof Setting q = e −π √ n in Theorem 4.1(ii)and employing the definition of J n , we arrive at (i). To prove (ii), we replace n by 1/n in part (i) and use Lemma 2.2.

respectively.
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