Some New Explicit Values of Quotients of Theta-Function 𝜙(𝑞) and Applications to Ramanujan's Continued Fractions

Abstract

We find some new explicit values of the parameter ℎ_𝑘,𝑛 for positive real numbers 𝑘 and 𝑛 involving Ramanujan's theta-function 𝜙(𝑞) and give some applications of these new values for the explicit evaluations of Ramanujan's continued fractions. In the process, we also establish two new identities for 𝜙(𝑞) by using modular equations.

1. Introduction

For 𝑞∶=𝑒^{2𝜋𝑖𝑧}, Im(𝑧)>0, define Ramanujan's theta-function 𝜙(𝑞) as 𝜙𝑞∶=^∞_𝑛=−∞𝑞^𝑛2=𝜗₃(0,2𝑧), (1) where 𝜗₃ [1, page 464] is one of the classical theta-functions.

In his notebook [2, volume I, page 248], Ramanujan recorded several explicit values of theta-functions 𝜙(𝑞) and its quotients which are proved by Berndt and Chan [3]. They also found some new explicit values. An account of these can also be found in Berndt's book [4]. Yi [5] also evaluated many new values of 𝜙(𝑞) by finding explicit values of the parameters ℎ_𝑘,𝑛 and ℎ^′_𝑘,𝑛 for positive real numbers 𝑘 and 𝑛 which are defined by ℎ_𝑘,𝑛𝜙𝑞∶=𝑘^1/4𝜙𝑞^𝑘,𝑞=𝑒^{−𝜋𝑛/𝑘},ℎ^′_𝑘,𝑛𝜙∶=−𝑞𝑘^1/4𝜙−𝑞^𝑘,𝑞=𝑒^{−2𝜋𝑛/𝑘}. (2) Yi [5] established several properties of these parameters and found their explicit values by appealing to transformation formulas and theta-function identities for 𝜙(𝑞). Recently, Saikia [6] found many new explicit values of quotients of 𝜙(𝑞) by finding explicit values of the parameter 𝐴_𝑛 which is a particular case of the parameter ℎ_𝑘,𝑛 where 𝑘=4. Saikia [6] also established some new theta-function identities for 𝜙(𝑞).

In the sequel of the previous work, in this paper we find some new explicit values of the parameters ℎ_4,𝑛 and ℎ_2,𝑛 which are particular cases of the parameter ℎ_𝑘,𝑛 by using some properties of ℎ_𝑘,𝑛 established in [5] and two new theta-function identities for 𝜙(𝑞). In addition, we give some applications of these new values of ℎ_𝑘,𝑛 for the explicit evaluations of Ramanujan's continued fractions 𝑐(𝑞) and 𝐾(𝑞) defined by 𝑐𝑞1∶=1₊2𝑞1−𝑞²₊𝑞²(1+𝑞²)²1−𝑞⁶₊𝑞⁴(1+𝑞⁴)²1−𝑞¹⁰_+⋯,𝑞<1,𝐾𝑞𝑞∶=^1/21+𝑞₊𝑞²1+𝑞³₊𝑞⁴1+𝑞⁵_+⋯,𝑞<1. (3)(4) The continued fraction 𝑐(𝑞) is studied by Adiga and Anitha [7]. For explicit evaluations of 𝑐(𝑞), see [6]. The continued fraction 𝐾(𝑞) is called Ramanujan-Gö‌llnitz-Gordon continued fraction [4, page 50]. For further references on 𝐾(𝑞), see [8, 9].

In Section 2, we record some preliminary results. Section 3 is devoted to prove two new identities for theta-function 𝜙(𝑞). In Section 4, we find some new explicit values of the parameter ℎ_4,𝑛. In Section 5, we evaluate some new values of the parameter ℎ_2,𝑛. Finally in Section 6, we give applications of these values of ℎ_4,𝑛 and ℎ_2,𝑛 for the explicit evaluations of Ramanujan's continued fractions 𝑐(𝑞) and 𝐾(𝑞).

We end the introduction by defining Ramanujan's modular equation. The complete elliptic integral of the first kind 𝐾(𝑘) is defined by 𝐾(𝑘)∶=₀^𝜋/2𝑑𝜙1−𝑘²sin²𝜙=𝜋2^∞_𝑛=0(1/2)²_𝑛(𝑛!)²𝑘^2𝑛=𝜋2²𝐹₁12,12;1;𝑘², (5) where 0<𝑘<1, ₂𝐹₁ denotes the ordinary or Gaussian hypergeometric function, and (𝑎)_𝑛=𝑎(𝑎+1)(𝑎+2)⋯(𝑎+𝑛−1). (6) The number 𝑘 is called the modulus of 𝐾, and 𝑘^′∶=1−𝑘² is called the complementary modulus. Let 𝐾,𝐾^′,𝐿, and 𝐿^′ denote the complete elliptic integrals of the first kind associated with the moduli 𝑘, 𝑘^′, 𝑙, and 𝑙^′, respectively. Suppose that the equality 𝑛𝐾^′𝐾=𝐿^′𝐿 (7) holds for some positive integer 𝑛. Then a modular equation of degree 𝑛 is a relation between the moduli 𝑘 and 𝑙 which is implied by (7).

If we set 𝐾𝑞=exp−𝜋^′𝐾,𝑞^′𝐿=exp−𝜋^′𝐿, (8) we see that (7) is equivalent to the relation 𝑞^𝑛=𝑞^′. Thus, a modular equation can be viewed as an identity involving theta-functions at the arguments 𝑞 and 𝑞^𝑛. Ramanujan recorded his modular equations in terms of 𝛼 and 𝛽, where 𝛼=𝑘² and 𝛽=𝑙². We say that 𝛽 has degree 𝑛 over 𝛼. The multiplier 𝑚 connecting 𝛼 and 𝛽 is defined by 𝑚=𝑧₁/𝑧_𝑛 where 𝑧_𝑟=𝜙²(𝑞^𝑟).

2. Preliminary Results

Lemma 1 (see [5, page 385, Theorem 2.2]). For positive real numbers 𝑘 and 𝑛, (i)ℎ_𝑘,1=1, (ii)ℎ_𝑘,1/𝑛=1/ℎ_𝑘,𝑛.

Lemma 2 (see [5, page 387, Corollary 2.6]). For positive real numbers 𝑘 and 𝑛, ℎ_𝑘²_,𝑛=ℎ_{𝑘,𝑛𝑘}ℎ_{𝑘,𝑛/𝑘}. (9)

Lemma 3 (see [5, page 392, Theorem 4.6]). One has √2ℎ_2,2𝑛ℎ_2,4𝑛+1ℎ_2,2𝑛ℎ_2,4𝑛=ℎ_2,4𝑛ℎ_2,𝑛+2, (10) for any positive real number n.

Lemma 4 (see [10, page 122, Entry 10 (i) and (v)]). One has(i)𝜙(𝑞)=√𝑧, (ii)𝜙(𝑞⁴)=√𝑧(1+(1−𝛼)^1/4))/2.

Lemma 5 (see [10, page 280-281, Entry 13(vii)]). If 𝛽 has degree 5 over 𝛼, then 𝛼𝛽^31/8+(1−𝛼)1−𝛽^31/8=𝛼³𝛽^1/8+(1−𝛼)³(1−𝛽)^1/8. (11)

Lemma 6 (see [10, page 314, Entry 19(i)]). If 𝛽 has degree 7 over 𝛼, then 𝛼𝛽^1/8+(1−𝛼)(1−𝛽)^1/8=1. (12)

3. Two New Identities for 𝜙(𝑞)

In this section, we prove two new identities for theta-function 𝜙(𝑞) by using Ramanujan's modular equations and transformation formulas.

Theorem 7. If 𝑃=𝜙(𝑞)/𝜙(𝑞⁴), 𝑄=𝜙(𝑞⁵)/𝜙(𝑞²⁰), then 𝑃⁶−256𝑃𝑄+640𝑃²𝑄−640𝑃³𝑄+320𝑃⁴𝑄−70𝑃⁵𝑄+640𝑃𝑄²−1600𝑃²𝑄²+1600𝑃³𝑄²−785𝑃⁴𝑄²+160𝑃⁵𝑄²−640𝑃𝑄³+1600𝑃²𝑄³−1620𝑃³𝑄³+800𝑃⁴𝑄³−160𝑃⁵𝑄³+320𝑃𝑄⁴−785𝑃²𝑄⁴+800𝑃³𝑄⁴−400𝑃⁴𝑄⁴+80𝑃⁵𝑄⁴−70𝑃𝑄⁵+160𝑃²𝑄⁵−160𝑃³𝑄⁵+80𝑃⁴𝑄⁵−16𝑃⁵𝑄⁵+𝑄⁶=0. (13)

Proof. Transcribing 𝑃 and 𝑄 using Lemma 4(i) and (iv) and then simplifying, we get (1−𝛼)^1/4=2𝑃−1,1−𝛽^1/4=2𝑄−1, (14) where 𝛽 has degree 5 over 𝛼.
Equivalently, 2𝛼=1−𝑃−1⁴2,𝛽=1−𝑄−1⁴. (15) Now by Lemma 5, we have 𝛼𝛽^31/8−𝛼³𝛽^1/8=(1−𝛼)³(1−𝛽)^1/8−(1−𝛼)1−𝛽^31/8. (16) Squaring (16) and simplifying, we arrive at 𝛼𝛽^31/4+𝛼³𝛽^1/4=𝑥+2𝛼𝛽^1/2, (17) where𝑥=(1−𝛼)³1−𝛽^1/4+(1−𝛼)1−𝛽^31/4−2(1−𝛼)1−𝛽^1/2. (18) Squaring (17) and simplifying, we obtain 𝛼𝛽^31/2+𝛼³𝛽^1/2=𝑦+4𝑥𝛼𝛽^1/2, (19) where 𝑦=𝑥²+2𝛼𝛽.
Squaring (19) and simplifying, we obtain 𝛼𝛽³+𝛼³𝛽+2𝛼²𝛽²−𝑦²−16𝑥²𝛼𝛽=8𝑥𝑦𝛼𝛽^1/2. (20) Again squaring (20), we obtain 𝛼𝛽³+𝛼³𝛽+2𝛼²𝛽²−𝑦²−16𝑥²𝛼𝛽²−64𝑥²𝑦²𝛼𝛽=0. (21) Now employing (14) and (15) and factorizing using Mathematica, we deduce that 𝑓(𝑃,𝑄)𝑔(𝑃,𝑄)ℎ(𝑃,𝑄)=0, (22) where 𝑓(𝑃,𝑄)=(𝑃−𝑄)⁴,𝑔(𝑃,𝑄)=𝑃⁶−256𝑃𝑄+640𝑃²𝑄−640𝑃³𝑄+320𝑃⁴𝑄−70𝑃⁵𝑄+640𝑃𝑄²−1600𝑃²𝑄²+1600𝑃³𝑄²−785𝑃⁴𝑄²+160𝑃⁵𝑄²−640𝑃𝑄³+1600𝑃²𝑄³−1620𝑃³𝑄³+800𝑃⁴𝑄³−160𝑃⁵𝑄³+320𝑃𝑄⁴−785𝑃²𝑄⁴+800𝑃³𝑄⁴−400𝑃⁴𝑄⁴+80𝑃⁵𝑄⁴−70𝑃𝑄⁵+160𝑃²𝑄⁵−160𝑃³𝑄⁵+80𝑃⁴𝑄⁵−16𝑃⁵𝑄⁵+𝑄⁶,ℎ(𝑃,𝑄)=16𝑃¹⁰−96𝑃⁹𝑄−32𝑃¹⁰𝑄+240𝑃⁸𝑄²+192𝑃⁹𝑄²+24𝑃¹⁰𝑄²−320𝑃⁷𝑄³−480𝑃⁸𝑄³−144𝑃⁹𝑄³−8𝑃¹⁰𝑄³+256𝑃⁶𝑄⁴+608𝑃⁷𝑄⁴+384𝑃⁸𝑄⁴+40𝑃⁹𝑄⁴+𝑃¹⁰𝑄⁴−448𝑃⁵𝑄⁵+352𝑃⁶𝑄⁵−1264𝑃⁷𝑄⁵+248𝑃⁸𝑄⁵−70𝑃⁹𝑄⁵+256𝑃⁴𝑄⁶+352𝑃⁵𝑄⁶−880𝑃⁶𝑄⁶+1640𝑃⁷𝑄⁶−785𝑃⁸𝑄⁶+160𝑃⁹𝑄⁶−320𝑃³𝑄⁷+608𝑃⁴𝑄⁷−1264𝑃⁵𝑄⁷+1640𝑃⁶𝑄⁷−1620𝑃⁷𝑄⁷+800𝑃⁸𝑄⁷−160𝑃⁹𝑄⁷+240𝑃²𝑄⁸−480𝑃³𝑄⁸+384𝑃⁴𝑄⁸+248𝑃⁵𝑄⁸−785𝑃⁶𝑄⁸+800𝑃⁷𝑄⁸−400𝑃⁸𝑄⁸+80𝑃⁹𝑄⁸−96𝑃𝑄⁹+192𝑃²𝑄⁹−144𝑃³𝑄⁹+40𝑃⁴𝑄⁹−70𝑃⁵𝑄⁹+160𝑃⁶𝑄⁹−160𝑃⁷𝑄⁹+80𝑃⁸𝑄⁹−16𝑃⁹𝑄⁹+16𝑄¹⁰−32𝑃𝑄¹⁰+24𝑃²𝑄¹⁰−8𝑃³𝑄¹⁰+𝑃⁴𝑄¹⁰. (23) By examining the behavior of the first factor 𝑓(𝑃,𝑄) and the last factor ℎ(𝑃,𝑄) of the left-hand side of (22) near 𝑞=0, it can be seen that there is a neighborhood about the origin, where these factors are not zero. Then the second factor 𝑔(𝑃,𝑄) is zero in this neighborhood. By the identity theorem, this factor is identically zero. Hence, we complete the proof.

Theorem 8. If 𝑃=𝜙(𝑞)/𝜙(𝑞⁴), 𝑄=𝜙(𝑞⁷)/𝜙(𝑞²⁸), then 𝑃⁸−4096𝑃𝑄+14336𝑃²𝑄−21504𝑃³𝑄+17920𝑃⁴𝑄−8736𝑃⁵𝑄+2352𝑃⁶𝑄−280𝑃⁷𝑄+14336𝑃𝑄²−51968𝑃²𝑄²+80640𝑃³𝑄²−69440𝑃⁴𝑄²+35056𝑃⁵𝑄²−9772𝑃⁶𝑄²+1176𝑃⁷𝑄²−21504𝑃𝑄³+80640𝑃²𝑄³−129472𝑃³𝑄³+115360𝑃⁴𝑄³−60424𝑃⁵𝑄³+17528𝑃⁶𝑄³−2184𝑃⁷𝑄³+17920𝑃𝑄⁴−69440𝑃²𝑄⁴+115360𝑃³𝑄⁴−106330𝑃⁴𝑄⁴+57680𝑃⁵𝑄⁴−17360𝑃⁶𝑄⁴+2240𝑃⁷𝑄⁴−8736𝑃𝑄⁵+35056𝑃²𝑄⁵−60424𝑃³𝑄⁵+57680𝑃⁴𝑄⁵−32368𝑃⁵𝑄⁵+10080𝑃⁶𝑄⁵−1344𝑃⁷𝑄⁵+2352𝑃𝑄⁶−9772𝑃²𝑄⁶+17528𝑃³𝑄⁶−17360𝑃⁴𝑄⁶+10080𝑃⁵𝑄⁶−3248𝑃⁶𝑄⁶+448𝑃⁷𝑄⁶−280𝑃𝑄⁷+1176𝑃²𝑄⁷−2184𝑃³𝑄⁷+2240𝑃⁴𝑄⁷−1344𝑃⁵𝑄⁷+448𝑃⁶𝑄⁷−64𝑃⁷𝑄⁷+𝑄⁸=0. (24)

Proof. Transcribing 𝑃 and 𝑄 using Lemma 4(i) and (iv) and then simplifying, we obtain (1−𝛼)^1/4=2𝑃−1,1−𝛽^1/4=2𝑄−1, (25) where 𝛽 has degree 7 over 𝛼.
Equivalently, 2𝛼=1−𝑃−1⁴2,𝛽=1−𝑄−1⁴. (26) Now by Lemma 6, we have (1−𝛼)1−𝛽^1/8=1−𝛼𝛽^1/8. (27) Squaring (27) and simplifying, we obtain 𝑥−𝛼𝛽^1/4=−2𝛼𝛽^1/8, (28) where 𝑥={(1−𝛼)(1−𝛽)}^1/4−1.
Squaring (28) and simplifying, we obtain 𝑥²+𝛼𝛽^1/2=(4+2𝑥)𝛼𝛽^1/4. (29) Squaring (29) and simplifying, we obtain 𝑥⁴+𝛼𝛽=(4+2𝑥)²−2𝑥²𝛼𝛽^1/2. (30) Again, squaring (30), we arrive at 𝑥⁴+𝛼𝛽²=(4+2𝑥)²−2𝑥²²𝛼𝛽. (31) Employing (25) and (26) in (31) and simplifying with the help of Mathematica, we complete the proof.

4. New Values of ℎ_4,𝑛

In this section, we find some new values of ℎ_4,𝑛 and ℎ_2,𝑛 by using theta-function identities proved in Section 3 and the properties of ℎ_𝑘,𝑛 listed in Lemmas 1 and 2. We begin with following remarks.

Remark 9. The values of ℎ_𝑘,𝑛 are real and ℎ_𝑘,𝑛<1 for all 𝑛>1 if 𝑘>1. We also note that the values of ℎ_𝑘,𝑛 decrease as 𝑛 increases when 𝑘>1. In view of this, in the following theorem we have ℎ_4,4>ℎ_4,5>ℎ_4,7>ℎ_4,8>ℎ_4,25>ℎ_4,49 where ℎ_4,4 and ℎ_4,8 are evaluated in [6, Theorem 4.3]. Yi [5] also evaluated the value of ℎ_4,4.

Theorem 10. One has(i)ℎ_4,5=(22+10√5−√2−√10−(√2+√10−22+10√5)²−4)/2, (ii)ℎ_4,1/5=(22+10√5−√2−√10+(√2+√10−22+10√5)²−4)/2, (iii)ℎ_4,25=114−80√2+25965−18360√2−(114−80√2+25965−18360√2)²−1, (iv)ℎ_4,1/25=114−80√2+25965−18360√2+(114−80√2+25965−18360√2)²−1, (v)ℎ_4,7=(12−7√2−7(34−24√2))/2, (vi)ℎ_4,1/7=(12−7√2+7(34−24√2))/2, (vii)ℎ_4,49=𝑎+√𝑏−−1+(𝑎+√𝑏)², (viii)ℎ_4,1/49=𝑎+√𝑏+−1+(𝑎+√𝑏)²,
 where 𝑎=1317−931√2+3470684−2454144√2 and 𝑏 = 6940535 − 4907700√2 + 13061331967/(123953−87648√2) − 92357564014/(123953−87648√2).

Proof. For (i) and (ii), setting 𝑘=4 and employing the definition of ℎ_𝑘,𝑛 in Theorem 7, we get 𝑃=√2ℎ_4,𝑛,𝑄=√2ℎ_4,25𝑛. (32) Setting 𝑛=1/5 in (32), applying to (13), and then simplifying using Lemma 1(ii), we obtain ℎ⁶_4,5+ℎ⁻⁶_4,5−70ℎ⁴_4,5+ℎ⁻⁴_4,5+320√2ℎ³_4,5+ℎ⁻³_4,5−1425ℎ²_4,5+ℎ⁻²_4,5+1920√2ℎ_4,5+ℎ⁻¹_4,5−3348=0. (33) Equivalently, 𝐴⁶−76𝐴⁴+320√2𝐴³−1136𝐴²+960√2𝐴−640=0, (34) where 𝐴∶=ℎ_4,5+ℎ⁻¹_4,5. (35) Solving (34) for 𝐴 and noting that 𝐴 has positive real value greater than 1, we obtain 𝐴=−√2−√10+22+10√5. (36) Invoking (36) in (35), solving for ℎ_4,5, and using the fact in Remark 9, we complete the proof of (i). Noting ℎ_4,1/5=1/ℎ_4,5 from Lemma 1(ii), we arrive at (ii).
For proofs of (iii) and (iv), we set 𝑛=1 in (32), applying to (13) and then simplifying using Lemma 1(i), we arrive at ℎ²_4,25+ℎ⁻²_4,25−456−320√2ℎ_4,25+ℎ⁻¹_4,25−674−840√2=0. (37)
Equivalently, 𝐵²+8−57+40√2𝐵+−674+840√2=0, (38) where 𝐵∶=ℎ_4,25+ℎ⁻¹_4,25. (39) Solving (38) for 𝐵 and noting that 𝐵 has positive real value greater than 1, we obtain 𝐵=2114−80√2+55193−3672√2. (40) Invoking (40) in (39), solving for ℎ_4,25, and using the fact in Remark 9, we complete the proof of (iii). Noting ℎ_4,1/25=1/ℎ_4,25 from Lemma 1(iv), we prove (ii).
To prove (v) and (vi), applying the definition of ℎ_4,𝑛 in Theorem 8, we get 𝑃=√2ℎ_4,𝑛,𝑄=√2ℎ_4,49𝑛. (41) Setting 𝑛=1/7 in (41), applying in (24), and then simplifying using Lemma 1(ii), we arrive at ℎ¹⁶_4,7−280ℎ¹⁴_4,7+2352√2ℎ¹³_4,7−18508ℎ¹²_4,7+44016√2ℎ¹¹_4,7−140616ℎ¹⁰_4,7+159264√2ℎ⁹_4,7−262810ℎ⁸_4,7+159264√2ℎ⁷_4,7−140616ℎ⁶_4,7+44016√2ℎ⁵_4,7−18508ℎ⁴_4,7+2352√2ℎ³_4,7−280ℎ²_4,7+1=0. (42) Dividing (42) by ℎ⁸_4,7 and simplifying, we get ℎ⁸_4,7+ℎ⁻⁸_4,7−280ℎ⁶_4,7+ℎ⁻⁶_4,7+2352√2ℎ⁵_4,7+ℎ⁻⁵_4,7−18508ℎ⁴_4,7+ℎ⁻⁴_4,7+44016√2ℎ³_4,7+ℎ⁻³_4,7−140616ℎ²_4,7+ℎ⁻²_4,7+159264√2ℎ_4,7+ℎ⁻¹_4,7−262810=0. (43) Equivalently, 𝐿⁸−288𝐿⁶+2352√2𝐿⁵−16808𝐿⁴+32256√2𝐿³−69120𝐿²+38976√2𝐿−18032=0, (44) where 𝐿∶=ℎ_4,7+ℎ⁻¹_4,7. (45) Solving (44) by using Mathematica and noting that 𝐿 has positive real value greater that 1 satisfying the fact in Remark 9, we obtain 𝐿=12−7√2. (46) Employing (46) in (45), solving for ℎ_4,7, and using the fact in Remark 9, we complete the proof of (v). Noting ℎ_4,1/7=1/ℎ_4,7, we arrive at (vi).
For proofs of (vii) and (viii), setting 𝑛=1 in (41), applying in (24), and simplifying using Lemma 1(ii), we arrive at ℎ⁴_4,49+ℎ⁻⁴_4,49+4−1317+931√2ℎ³_4,49+ℎ⁻³_4,49+28−2053+1452√2ℎ²_4,49+ℎ⁻²_4,49+56−2409+1703√2ℎ_4,49+ℎ⁻¹_4,49+210−837+592√2=0. (47) Equivalently,𝐷⁴+8−1317+931√2𝐷³+16−3593+2541√2𝐷²+64−1614+1141√2𝐷+128−475+336√2=0, (48) where 𝐷=ℎ_4,49+ℎ⁻¹_4,49. (49) Solving (48) for 𝐷 and noting that 𝐷 has positive real value greater than 1, we obtain 𝐷=2𝑎+√𝑏, (50) where 𝑎=1317−931√2+3470684−2454144√2 and 𝑏 = 6940535 − 4907700√2 + 13061331967/(123953−87648√2) − 92357564014/(123953−87648√2).
Invoking (50) in (49), solving for ℎ_4,49, and using the fact in Remark 9, we arrive at (vi). Noting ℎ_4,1/49=1/ℎ_4,49 from Lemma 1(ii), we complete the proof of (vii).

5. New Values of ℎ_2,𝑛

In this section, we find some new values of the parameter ℎ_2,𝑛 by using the values of ℎ_4,𝑛 evaluated in Section 4 and in [6].

Theorem 11. One has(i)ℎ_2,6=8√2−10+(√2−2)18−12√2, (ii)ℎ_2,3/2=(2−√2+18−12√2)/(28√2−10+(√2−2)18−12√2), (iii)ℎ_1/6=1/8√2−10+(√2−2)18−12√2, (iv)ℎ_2,2/3=28√2−10+(√2−2)18−12√2/(2−√2+18−12√2).

Proof. Setting 𝑘=2 and 𝑛=3 in Lemma 2, we deduce that ℎ_4,3=ℎ_2,6ℎ_2,3/2. (51) From [6, page 174, Theorem 4.3(vii)], we have ℎ_4,3=2−√2+18−12√22. (52) Combining (51) and (52), we obtain ℎ_2,6ℎ_2,3/2=2−√2+18−12√22. (53) Next, setting 𝑛=1/6 in Lemma 3 and simplifying using Lemma 1(ii), we obtain √2ℎ_2,6ℎ_2,3/2+ℎ_2,6ℎ_2,3/2⁻¹=ℎ_2,6ℎ_2,3/2+2. (54) Invoking (53) in (54) and simplifying, we deduce that ℎ_2,6ℎ_2,3/2=28√2−10+√2−218−12√22−√2+18−12√2. (55) Multiplying (53) and (55) and simplifying, we complete the proof of (i). Dividing (53) by (55) and simplifying, we arrive at (ii). (iii) and (iv) follow from (i) and (ii), respectively, and Lemma 1(ii).

The proofs of Theorems 12–15 are identical to the proof of Theorem 11. So we omit details and give only references of the required results to prove them.

Theorem 12. One has(i)ℎ_2,10=(4+𝑐²)√2−4𝑐/2, (ii)ℎ_2,5/2=𝑐/(4+𝑐²)√2−4𝑐, (iii)ℎ_2,1/10=2/(4+𝑐²)√2−4𝑐, (iv)ℎ_2,2/5=(4+𝑐²)√2−4𝑐/𝑐,
 where 𝑐=22+10√5−√2−√10−(√2+√10−22+10√5)²−4.

Proof of Theorem 12 follows from Theorem 10(i), Lemma 2 with 𝑘=2 and 𝑛=5, Lemma 1(ii), and Lemma 3 with 𝑛=1/10.

Theorem 13. One has(i)ℎ_2,50=−2𝑑+√2(𝑑²+1), (ii)ℎ_2,25/2=𝑑/−2𝑑+√2(𝑑²+1), (iii)ℎ_2,1/50=1/−2𝑑+√2(𝑑²+1), (iv)ℎ_2,2/25=−2𝑑+√2(𝑑²+1)/𝑑,
   where 𝑑=114−80√2+25965−18360√2−(114−80√2+25965−18360√2)²−1.

To prove Theorem 13, we use Theorem 10(iii), Lemma 2 with 𝑘=2 and 𝑛=25, Lemma 1(ii), and Lemma 3 with 𝑛=1/50.

Theorem 14. One has(i)ℎ_2,14=(−8+6√2)(12−7√2−238−168√2), (ii)ℎ_2,7/2=12−7√2−238−168√2/2−8+6√2, (iii)ℎ_2,1/14=1/(−8+6√2)(12−7√2−238−168√2), (iv)ℎ_2,2/7=2−8+6√2/12−7√2−238−168√2.

We employ Theorem 10(v), Lemma 2 with 𝑘=2 and 𝑛=7, Lemma 1(ii), and Lemma 3 with 𝑛=1/14 to prove Theorem 14.

Theorem 15. One has(i)ℎ_2,98=2(𝑎+√𝑏−(𝑎+√𝑏)²−1)(√2𝑎+√2𝑏−1), (ii)ℎ_2,49/2=(𝑎+√𝑏−(𝑎+√𝑏)²−1)/2(√2𝑎+√2𝑏−1), (iii)ℎ_2,1/98=1/2(𝑎+√𝑏−(𝑎+√𝑏)²−1)(√2𝑎+√2𝑏−1), (iv)ℎ_2,2/49=2(√2𝑎+√2𝑏−1)/(𝑎+√𝑏−(𝑎+√𝑏)²−1),
 where a and b are given in Theorem 10(viii).

Proof follows from Theorem 10(vii), Lemma 2 with 𝑘=2 and 𝑛=49, Lemma 1(ii), and Lemma 3 with 𝑛=1/98.

6. Applications of ℎ_4,𝑛 and ℎ_2,𝑛

In this section, we use the new values of the parameters ℎ_4,𝑛 and ℎ_2,𝑛 to find explicit values of continued fractions 𝑐(𝑞) and 𝐾(𝑞) defined in (3) and (4), respectively.

The parameter ℎ_4,𝑛 is useful in finding explicit values of the continued fraction 𝑐(𝑞). If we know values of the parameter ℎ_4,𝑛 for any positive real number 𝑛, then explicit values of 𝑐(𝑒^{√−𝜋𝑛/2}) can be calculated by appealing to the following theorem.

Theorem 16 (see [6, page 177, Theorem 5.1]). One has 𝑐𝑒^{√−𝜋𝑛/2}=1√2ℎ_4,𝑛, (56) where 𝑛 is any positive real number.

For example, employing the value of ℎ_4,5 from Theorem 10(i) in Theorem 16, we obtain 𝑐𝑒^{√−𝜋5/2}=√2×22+10√5−√2−√−10√2+√10−22+10√5²−4⁻¹. (57) Similarly, we can find new values of 𝑐(𝑒^{√−𝜋/25}), 𝑐(𝑒^−5𝜋/2), 𝑐(𝑒^−𝜋/10), 𝑐(𝑒^{√−𝜋7/2}), 𝑐(𝑒^{√−𝜋/27}),  𝑐(𝑒^−7𝜋/2), and 𝑐(𝑒^−𝜋/14) by employing the values of ℎ_4,𝑛 from Theorem 10(ii)–(viii), respectively, in Theorem 16. Since it is a routine calculation, we omit details.

Next, the parameter ℎ_2,𝑛 is connected to continued fraction 𝐾(𝑞) by the following theorem.

Theorem 17 (see [8, page 281, Theorem 4.1]). For any positive real number n, one has 𝐾²𝑒^{−𝜋𝑛/2}=2^1/4ℎ_2,𝑛−12^1/4ℎ_2,𝑛.+1 (58)

From Theorem 17, we note that if the values of ℎ_2,𝑛 are known, then the values of 𝐾(𝑒^{−𝜋𝑛/2}) can easily be evaluated. For example, using the value of ℎ_2,6 from Theorem 11(i) in Theorem 17, we evaluate 𝐾𝑒^−𝜋6/2=2^1/48√2−10+√2−218−12√2−12^1/48√2−10+√2−218−12√.2+1 (59)

Similarly, we can evaluate new values of 𝐾(𝑒^{−𝜋𝑛/2}) for 𝑛 = 3/2, 1/6, 2/3, 10, 5/2, 1/10, 2/5, 50, 25/2, 1/50, 2/25, 14, 7/2, 1/14, 2/7, 98, 49/2, and 2/49 by using Theorem 17 and the values of ℎ_2,𝑛 evaluated in Theorems 11–15.

Acknowledgment

The author is thankful to the University Grants Commission, New Delhi, India for partially supporting the research work under the Grant no. F. No. 41-1394/2012(SR).