Certain New ScUafli T>Tpe Mixed Modular Equations

VieHiamJ.Maih. (2016) 44:501-513 DOI iO.I0O7/slO013-015-0162-y

Certain New ScUafli T>Tpe Mixed Modular Equations

Megadahalli S . M a b a d e v a N a i k a ' •

Sathyanarayana C h a n d a n k n m a r ^ - M a n j u n a t h H a r i s h '

Recen-ed: 20 May 2014 / Accepied: 21 March 2015 / Published online 8 Augua --015

e Vielnam Academy of Science and Technology (VAST, and Spnnger Scienc^+Bnsiness Media Singapore

Abstract Schlafl, (J. R e i n e Angew. M a t h . 72, 3 6 ( K 3 6 9 . 1870) has established modular equations i m - o h i n g kk and Xk' for degrees 3 . 5 , 7. 9. 1 1 . 1 3 , 17. and 19. On p a s e s 86 and 8 8 of his first Dotebook. Ramanujan recorded 11 Schlafli-type m o d u l a r equations for composite degrees. In this paper, w e establish several new Schlafli type mixed modular equations for composite degrees by elementary algebraic manipulations which are analogous to those recorded by Ramanujan.

Keywords Modular equations • Theta-functions

Mathematics Subject C l a s s i f i c a t i o n (2010) 3 3 D 1 0 - I I A 5 5 - 11F27 1 Introduction

Following Ramanujan. let ( a : 9 ) ^ denote the infmite product f { { l - aq") (a. q are complex n u m b e r s . i91 < I l a n d the Ramanujan t h e t a - f u n c t i o n / ( a ^ t ) ;

f(a.b):= £ ; «'.(«+l,,.2^„(„-i„2 \ab\<\.

'• Mi^adahalh S. Mahades'a Naika mimnaika^rediffmail com

Manjunath Han^h numhansh @ gmaiLo

Department of Ntathematics. Banialore UmversitN. Central College Campus

Bangalore. .S60 001 Kamataka. India ' Depanmem of M a ^ t ^ c s , M S. Ramaiah Unueism. of Applied Sciences, Peenva Campus

PBcnya4thPhase.Bengaluru.560058Kainataka.India

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related lo the above product by Jacob!'s fundamental factorization foimula fia, b) = l-a; ab)^{-b: ab)a,lab: ab)ap.

The following definitions ofdieta-fianctionsv with |9| < 1 ate classical:

(«(?) : = / ( « . « ) = f ; il"' = (-q:r,')i,(q':,,^-)^.

fl-q) := / ( - , . _ , 2 ) = g (^|)","U»-I)/2,^,^.^)_^

The ordmary or Gaussian hypetgeometric function is defined by 2Flla.b-,c:z>:=Yl la).(b).

o < W < 1, where a 7, A, and c are complex numbers such that c ^0,-i.-2 and

(0)0 = 1, (o), = o ( a + l ) . . . ( „ + „ - ! ) for any positive integer n.

„f i h T " ! " f *= " ° ' » " "' " " " " l " 'I™'*™- Let *r (t) be the complete eUiptic integral of die first kind of modulus A Recall diat

, 2 ^ InV

"U-^- ^m

modulus of i . h is classical to set qlk) =. e->"«r»'l/«r«) „ , , , , , . " '•™P»™oaiy from 0 to I. e i ? T O _ e ' so that j is one-lo-onc increasing

In the same manner inu-oduce Z. I = *rfl',l / ' — irrv, j

equality "uuLBi,, - / l u , ) , / . ^ -«(«,)andsupposeaialthefollowttig

«•' _ i'l

"' (f - Z7 (2) holds for some positive integer „,. Tlian a modular eqnadon of degree „, is an algebraic

anon between the mod.h , and . , which is induced by (2). F o l L i n g R ^ u S w s e t a . - ! ™<lfl-«,.>henwesay<Jisofde8ree„,o.er„.ThemultipI,er™isdefin«lby

m- ^ - ^^(g)

LI vHq'i) (3) for,? = e--''^'*'l/*^i*l,

k i n d c o m s L i ^ I ' ' • ' '•"' '"'• *•'• " ' ' ' = ' ' ™ " ' " " • ' ' ' " = " ' " P " ' ™"=8ials of the first Kino corresponding, m pairs, to the moduh./^ ,/S / u and / I o„,i ,u • moduli.respectively Let-, „ , ,„rt „ "^"•./''•.VJ'.aiKl Vi,andOieircomplementEiy dia, the equates ' ' " ' " " ™ ' " " e " " ' " '^"^ = » ! « • S»Ppo»

i'l K'

*^ i l - • " : ? - / . . „ ^3 (4)

S i r " " V T f ' ° ° ' ' " ' " ' " ' " ' " ' ° " ' ' " ""»"•'" '>=•»"" l-e moduli ^Jfjy.,i Va that IS induced by (4) We sav that fl ,. ,.„^ s r j V". VP-V>''^^^

c v e r « , T h e m u l t i p l i e i ; r ™ ^ ^ / ; ^ " V ^ ' ^ f ' ' - g ^ ' ' " . - « 2 . a n d « 3 , n ^ ^ ^ ^ )/,and3. ".//-lanam -/.2/Z.3 are algebraic relation involving a, ^,

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Certain New Schlafli Type Mixed Modular Equa

Jacobi and Sohncke have established modular equations involving c'''^ and fi'"^. Such modular equations are called Jacobi-Sohncke modular equations.

Russell [15] has obtained several modular equations of degrees n = 13. 17.29,43.53, and 59 involving (a^)''"* and {(I - ff)(l - ^)|l/8^ which are referred to as Russell type modular equations. For example

(o'^)''''' + | ( l - a ) ( l - ^ ) | ' / ^ = I . where fi is of degree 3 over a.

Schlafli has estdilished modular equations for degrees 3, 5, 7. 9, II. | 3 . 17, and 19 involving {4o'(l - a)]'^^^ and (4^(1 - ^ ) | l / ^ l TTiese modular equations are known as Schlafli type modular equations. Weber [18, p. 90] has also established irrational modular equation for composite degrees involving / . / ] , / . , and / j , where / = {4a(l - a ) ] ' / - " * / , = (4^(1 - ti)\''^\ f2 = {4>'(1 - Y)]^/^\ and / j = (4^(1 - S)\"''\ which are known as Weber type modular equations.

Ramanujan's class jnvanant G„ is defined by

Gn-=2-"^q-^P~'x(<!)- (5) where

X iq) = i-q\ qh^. q = exp(-7r Jii).

Since xiq) = 2'/^(«(l - e)/?)-'/^^ [3, Ch. 17, Entry 12(v). p. 124]. it follows from (5) that

t ^ „ = : i 4 a ( l - « ) | - i / 2 4 ,^, Recently. Mahadeva Naika [6] and Mahadeva Naika and Sushan Bairy [ 10] have obtained

several exphcil evaluations of class invanants using modular equations.

On pages 86 and 88 of his first notebook [14], Ramanujan recorded a total of 11 Schlafli-type modular equations for composite degrees. In 113], Ramanathan observed that one of diese 11 equations follows from a modular equation recorded by Ramanujan in Chapter 20 of second notebook. By using the dieory of modular forms. Bemdi (4] has proved these modular equations. In [1,2], Baniah proved these equations by deriving some theta-funclion identities from Schroter's formulae and using Ramanujan's theia-function identities. Mahadeva Naika and Bairy [11] have also established several new mixed modular equations in the form of Schlafli. For more details one can see [5, 7. 9, 12, 17].

Motivated by all these works, in diis paper, we establish nine new mixed modular equa- tions in the form of Schlafli We utilize .some identities involving ratios of theia-funciion in establishing these equations.

2 Preliminary Results

In this section, we list some

results. if die relevant identities which are useful in proving of c Lemma 1 [3, Ch. 17, Entry 12 (i) and (iii), p 124] ForO < .v < 1, we have

fie-') = ./E2-'/^{^il-x)e"]"'-\ (7, fie-'^)^^.2->^'\x{l-x)er''\ (8,

\^-herez:= 2^1 ( f ^ . \:x) andy := jt^^^^^,

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M. S Mahadeva Naika siLi Lemma 2 [3. Ch. 19, Entry 13 (xii) and (xiii), pp. 281-282) If^ is of degree 5 over a. that

ll>\"\(t-l>\"* lfli-Jl)\'"

Uj +lrr7j " I s n ^ j 'm- "«i

where m is the multiplier for degree 5.

' + i ' j - " ' n * ' + s j + ' « ' [ s + & j - 3 i 8 4 [ f + i ]

^ f + A ] (l360-24S[i + i ] + 176[fJ + g ] + 4 [ U + g]j+73l6=0.

Lemma 4 |8];/X := Jfl* oiid y := J * l , then

y^ 5y- i5y r 5 1 r . "i n ^2

1" + Jj + 33 [ y + J,] - 99 [y« + J,] + ,52, |-y3 + J^j -l683[y= + Jj+880o[y + i]=6534 + [ x ' + 4 ]

^ ' ' l [ ' ' ' r P ' ' + f]-[^-' + |i][.i+4[y4j,]] ,13)

- X = + | , l [ l 8 - 5 6 [ y + i l + 3[K2 + J , l - 8 [ y 3 + J,]]

- [ X + I ] [-126 [)' + f ]+ I60|y= +J,]-18 [,-= +J,]+324+9[y'+^]]|.

3 Mixed Modular Equations in the Form of Schliifli

In ihis section, we establish several new Schlalli type mixed modular equations. We sel

P := ( 2 5 6 « / J y « ( l - o ) ( l - « ( | _ y ) ( | _ j i „ l / M (14,

„.^lv.m-a)l\-S)\'m

XliyD-IDIl-y)) ('5) K .^ lyHl-y)ll-S)\'l"

U / S ( l - o ) ( l - / J ) J " « T •= / ' « y ( l - o ) ( l - y ) \ " »

\DSi\-ni\-S)) " "

Cenain New Schlafli Type Mixed Modular Equations

Tbeonm 1 If a. ^.y.andS have degmes 1.5,5, and 25, respeaively Ihen 16(7- + i ) + l , ( r 2 + J . ) + 6 ( T ' + 5L) + ( 7 - * + 5 L ) + 2 2

= (s= + i|,)[(r + i ) + (r^ + ^) + i].

Proof Equations (9) and (10) can be rewritten as

l « ( l - « ) l m\a{\-a)\ + ^•

EquMion (19) can be written as

(18)

where m := J ^ and fl •= I M=^

In a similar manner, we obtain

. h e r e „ ' : = t S 4 a . d i : = ( « l ^ r "

' " " ' 1 I ' d - I ' l I Employing (20) and (21) in (12). we deduce that

l)X-5yaub+5yauv-i3Xbva+iXbvu-%Xaub-Xauv-Xa-bi>

-Xaub'-lrXaubv+mym+2ybi,-tXbv-55yva-l-55yba-\0rbu - 4 2 1 ' - 2 6 y 6 + 8 y u - 2 y i , ! + 8 3 » - 2 9 J - » + 6 X f , = - 4 0 y i , - 5 y a - -l-Xd+iiXu+Xa'-Sya+iya^b-sya^v-myau + mxba-ATXbu

-•^Xva + nXm+l3Xb^a-^XI,'u-Xau-^SXa'b-^Xa\-l-XaV=a. (22) w h e r e i i ^ V l n - 1)^+ 20a and ii ^ ^ ( f c - l)2 + 20fc.

find^lr""'^ "'° " " " ' ' " " ' " " ' " B " ™ » " " * of (22) and Oien squaring both sides, we 6 2 4 - IMoiit'u - S21a'ub'v - 2la'ub'v + 132o=»i,u - I52ct'ub'-v +462<i-|,i,i, - I032oiit-i. - 45301,1,1, - ma'ub'v - a'ub'v + ya'abu -Ta'ub\ - 6500 + 255b'a'v + lOibVu + I6a'ub^ + )6a\b + )t,a*ub' -l-a'uv-l-a'b'u-l-a'b^ - 7o'i,„ + Ta'b'v - ma'bv + msbVv + 2 l 5 o V i , + 2556=uo-211,-'i,„ + 580i.'o=i,+ l 0 6 i V - l 3 7 0 a - ' l , i , +550lm,t- - Wia'bv - IOI3(,i™ + I6011B + 4330(,»<, + 94lo«fc +49l5o=»2|, + l6oJ|,i, + I78l<i\4-' + 2 7 6 o \ i + 2965*2„ + 274lo2„i,2 +396o',6= + 46o=«i. + 876<i'«4 - 3l7i-i,» + 34lo=«6-' + 30a'b\

+2906i>- + 62611 + 458 Id - 626« - IJOu^ - 1500^ - o ' - 25<,» + 6261,=

+614* + 24' - 540o«6' - 485o'(,= - 255l,«o' - 30a'b' -2b'v- I035ta -12770ta +46216,, +6OO00 - 624»« + 1601, - 35450^4 - ISOo^,, -208156=0 + 302662„ - 200900=62 - IdaV - l 6 o ' 6 ' + o<« - a'b' -a'v + 46o=n - 2720o-"6 - I50o'» - 56650=6= - 3l96=i, + I986oo6=

- l 0 1 3 5 o V -436=„ - 2556*0 + 216'„ - 5806«o= - 52604'o + 5066=,, (23) -42l04'o= + l 6 o \ -420o»6 - 25o-ii, - 16o-'6 + 21o=4''o =

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M. S. Mabadeva Naika et

Isolating the terms containmg i, on one side of (23) and then squaring both sides, we deduce 5804*o=iJ + 603730O V u + 954016' + 166866' + I4466» + 616' + 6'°

-I953I250 + 159267704=0=1, + 57344o'4=i, - 2048a=6» + 7l68o=4=n +56l85754'o=i, + o'6*ii + 52o=6'i, + 132906'»o= + 29609504=oo -595204't,o-' + 2900056'ou + 22977004'o=v + 13(M050o-'6''i, + 2554"o=u +304l5o=4'i, - 1435506'o-'u + I0186'uo= + 96089254''o=u - 25354'uo' +2556»oo + 257055o=6'i, + 92444=o\ + 24479S006''o=u + 1116l60o''6=u + I537l050o6*i> + 38245o=6S + 261120o'6u + .34020854=o=i, +39l025206=»o + 4286S4456=o=« + 138404'i,o + 593920o'6» + 30o''6«»

+ 195619704UO + 200842IOo=4=u + 446962856=uo + 1574245oVi, +112644= + 10244 - I024o= - 296964= + 1648644'' + 2389494' - 2 7 3 l 3 6 5 o V - 6229l0956*'o= - 9389695o-'4-* - 573444''B - 166864'o=

-41713954>o' + 5 6 7 0 3 0 6 V - 5l09l956»o - 100443456«o= - 92446'u -237602254'o= - 13591000*4= - 382456'u - 3437l600o6= - 9540loV -5950.39754=0= - 4043654'o - S54265o=4' - l446o=4' - 6lo=4' - o=6' -366465o=4' - 10184',, + 835354'o'' - l6l354»o - 35635o=4' - 526'i, -l55856«o= + 22654'o-' - 2554»o - 5S04'o= - 2556»o= - 300*4' - 6'|, + I0244» - 370786554O + l953l25i,o - 1692005o=6 - I503397354=o -342063350=6= + 29696o=4= - 164864o=4= - 238949o=4* - I l264o=4 -1024o=« + 5939200=4 + 4841 ll5o=4= + 20484=n - 2043909654=0 -1579122150=4= - 7l684=u - I204359706*o - 1623506456*0=

-514300154=0= + 26ll20o*4 + 3466240o*4= + 2214245o=6u = 0. (24)

Ehminaung i, from (24). we obtain

- 4 o - 30o=4 - 304=0 + 3194o=6= - 8810o=4= - 100590a'6= - 466770o'6*

-255o=6 - 255o=4 - 77IOo=4= - 2554=o - 77IOo=4= - 5804*o - I79106*o=

-586364=0= - 580o»4 - I79I0„*4= - I63290o*4= - 1632906*o= - 456866o*6*

-4667706'o= - 82304'o= - IO03354«„* + 304'o - 4667706=o' - I00590o'6=

+30o'4= - 8230o'4* + o'4 - I005906=o= - 466770o*6= - 8230o«4=

-100335o«4* + 30o»4 - 255o6= - 920122o'4' - 88104=o= + o« + 4»

-4568664'o' + 6'o - I63290„'6' + 30o=4' + „ ' V + 30o»6= - 8810o"4=

+o 6- + 30o'4* - 255o'6= - I79l0o»4' - 580o»6' + o»4= - 100590o=6' - I63290o'4' - 58636o'4' - 77IOo»4' - 255o'4' - 88I0„=4' - l79IOoV -77IOo'4" + 3l94o»4» - 30o'4» - 255o'4' - 82304'o* + o=6' + 306'o=

+4'o= + 30„'4' - 580o>4'' + o*6'» - 255o'4' - 30o«6' - o ' 6 ' = 0 (25)

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in New Schlafli Type Mixed Modular Equations ₅₀₇ Simplifying the above equation (25) after substimting for o and 6, we deduce that

(-e' + i6e=r= + r= - (2= - e* - g! - 2= + 6e'r= + i6e=r= - e'r' -B*z-'-i2=r« + e»7-=-e=7-'-e=r' + iie'r=+ 226*7'=+ iie=r=

+6r=e)(e'=+5e'=7-' - loe'r'+4e=r= + e'=7'=+e»7''=+g' +2-'-ios'7'» + 7os"r« + e'*r= + (2i<7'»-e"ri= + ioc»2-»

-(2'r'= - 7oe'r' - i5e'=r= - i5e'=7''+e' - fl" - (2= - o» + e ' -ioa'r= + i6e=r= + i6e"r= +1766*7' + 25e'*r' + 4ei=7-' -6oe"7'' - eoe'r' + 52*7' - lOfl'r' + i6fl=r' + I6B"r» - 6"?''=

+4ei=r' + fli'T-'+25e=7''+4e=7'»+e*r'=+5e'°r= - 7oe'7''

+ I0e»r= - 34e"7'' + 70(3=7'' - 34!5=7'« + 5e«J= - 150*7= - I5e*r'

+52'»r' + e'r'= - 6e'=r' - 687-' + e=r= + e=r' + 5fl'r') (e'= + 5!2'=r' - iofl'r= + 4e=7= + fli=7'= - fl»7''= - e' + r ' -I0(7'r» - 70(2"7'' - s»r= - e » r ' + Q"T'^- - loe'r' + Q'T"

+70i2'r' + l5!2'=r= + i5s'=r» + e* + e" + !2= - fl» - Q' - ioe'7=

+ 16(7=7= + l6g"T' + l76fl'T' +25e'*r' + 4(3"7= - e o g ' ^ r ' - 6 0 e ' 7 ' + 5(3*7' - I0!3'7' + I6e=7' + I 6 e " 7 ' - e=7'= + 4el=7' + fl"7' + 252=7' + 42=7' + 2*7" - 52'°7= + 702=7' - I02'7=

+342'=7' - 702=7' + 342=7' - 52'7= + 158*7= + 152*7' - 5 2 " 7 ' - e ' 7 ' = + 6e"7« + 627' - 2=7= - 2=7' - 52'7')(2' - 162=7=

+'•' - e ' + e * + e = - e ' - 6e'7= - i6s'7= + 2 ' 7 ' + 2 * 7 ' + 2 = 7 ' +2»7=-2=7-'-2=7' +ll2'7= + 222*7= +112=7=-67=2) = 0 (26) From 14, Ch. 34. pp.I89-l90]. we note that for « = 1/25 and us,ng die fact that C„ = tlltp. we have C, = I and C25 = t7i/25 = ^ . + ' .

Employing the values of C, and 62,. we "can compute die values of 0 and 7 then employing die resulting values of Q and 7 in (26), we find that the fim factor vanishes for 11 specific value of , = , - " « . Ihus by the identity theorem it vanishes identically D Theorem 2 If a. f. y. andShave degrees I. 5. 5. and 25 respectively then

( « = " - i ) - H ( . " + ^ ) - 2 9 ( « . = + J , ) - | « ( „ . + ± )

-259(.*+J,)-i6(7. + J , ) [ ( „ . + J , ) + ( . . , ^ ) „ ] +*(^^ + 7^) [(«'V-^)«(«>+J,)+4(„*+J,),4]=39„. (27)

Proof From [3. Entry 13 (xiv), Ch. 19. p. 282]. we have

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M. S. Mahadeva Naika etil;

w h e r e / l = (16ay3(l - ff)(l - ^ ) j ' ' " ^ and B = j ^ j j ^ ! "

Squanng both sides of (28). we get

Equation (29) can be written as

where C = (16yS(l - |/)(1 - J)ll/i= and D = Equation (31 lean be written as

(31)

(32)

..he„4.= J M = « | - „ , , , , [ , ( , _ , ) J =

Employing (30) and (32) in (25), we gel

378600," + 34762089600r* + 1349176960r',,= + .309760ir' + 515545120,i'r +222121751040.1,-= + I3339l9520i=/ + 13339l9520.!*r= - 7260j'r=

+S9440.i*r' + 73520r'.v= - 30j'f* - 90.=^" - 90j'r= - 9 0 » ' , ' -19544320.*,= + 1844560s'r= + 73520s'r= - 90.,'r' + 41034107520s*, -460r'i= + 4l034l07520r',< + 76188309120,=.!= + 22212l75IO40i=r +76188309120,=r= - 19544320,,=r* - 23l51640f's= - 30,*,' - 250i»,=

-842544!=,= + 900.!=,= + 2 3 0 0 s ' / - j ' r ' + 900s'r' - s ' , ' +515545120.!,' + I349176960s=r= + 1844560.!=,' - 23l51640s'r=

+2300.*," + 120982740480., + 14939378864.*,= + 9 0 0 . ' r ' + 20287200,,' -115072000.*,* + l3805769600()„-= + 138057696000.=, + 18408865664.=,=

+ I493937S8M,*.= + 78381,',* + 59398600,'.= - 10710,=.,= + 59398600,',=

+ IS7l843904OO.=r= - ,»,•• _ 842544,'.= + 37931,'.= - 250.=,' - .=,»

+78381.',* - 135526800.=,= - 135526800.=,= + 6393563744.', +639.3563744,=. + 24768320,' + 34762089600.* + 92817446400.=

+6I5390MO.' + 615390640,' + 378600.= + 6536275200.' + 24768320.' +6536275200,= + 92817446400,= + 1160,' + 100818950400,=

+ 100818950400.= + 1,60,' - 460.',= + 5 0 6 , , ' + 20287200.', + 89440,',*

-7260,-V' + 37931 !*,= + 309760s', + 5 0 6 . ' , - sV + . " ' + r ' " = 0 (33) 0 Spnnger

Certam New ScMafli Type Mixed Modular Equauons SQ.

Substimthig for , , J and then settuig /IC = P and i = «, we deduce dial (1 + 807= «* + 807««» + SOP'R' + I60««7* + 80J>*ff= + 1 6 0 / > ' B * - 2 5 9 f ' R ' - 16R'P' + K'Of'O - 259?* fi* - 29P=S= - 167'fl* - 297"««

- 1 6 P = « ' - I67'«= - I 4 P ' R » - 1 6 7 ' « ' + 4 0 7 ' / ; ' - 16P'7(= + I 6 0 7 ' f i ' + 1 6 0 7 ' « ' - 1 4 4 7 ' S ' - 1447=«= - 390«=p' - I 6 R ' 7 = - 16P'R' + I607=«' + l 6 0 7 ' R = - l 6 7 = « - l 6 R ' 7 + 40P'7i' + 4 0 7 = f l - l 4 7 « + 4 0 « = 7 ) (i + 807= «* + 8 0 7 ' R " + 8 0 7 ' f i ' + I 6 0 R ' 7 * + 807*«= + Iftop's*

- 2 5 9 7 ' K ' - 16«"7* + «'»7i» _ 2597*7(* - 297=S= - 167= S* - 297»/(«

- 1 6 7 = « ' - I67'«= + MP'R' + |67»R= - 4 0 7 » R ' + 167'R= - 1607'«=

-I607=R' + I 4 4 P ' « ' + I44P=R= + 390R=P' + 16«'p= + I6P=«' - I 6 0 P = R ' - 1 6 0 P ' R = + 1 6 P ' R + I 6 R ' P - 4 0 p ' f i ' - 40P=R + H P ) ! - 40/(=P) ( - I 6 P = R * - I 6 P ' R " - 16P'S« - 29P»S= - 29«''p= - 259)!«p* - 16P*R=

- 2 5 9 7 ' R * + I60P'S» + 80R'P< + I60P*«* + 8 0 P ' s * + 807=«' + 8 0 P ' R = + 7 ' » + fl«' + 4 0 P ' » + 4 0 P / ( ' - K P ' R - 16P»S=- I447'ff=+ I 6 0 P ' R ' + 1607=«' - I 6 P ' ) ( ' - l6P=fi' - 390R=P' - I 4 4 R ' P = + 40P=R» - 167= p ' + I 6 0 P = P = + 1607= R= - I6P=« - 16R=P - l4Pfl' + 4 0 7 ' R = | (-I6P=»* - I 6 P ' R ' - I 6 P ' R ' - 297«fi= - 29«V= - 259R'p* - I67*P=

- 2 5 9 7 ' R * + l e O P ' p ' + SOR'P* + I60P*)!' + 80P»P* + 8 0 P = R ' + 8 0 P ' R = + P ' » + «'» - 407'fi - 407R' + 147»R + I 6 7 » R ' + I44p'p= - l 6 0 P ' f i ' - 1 6 0 p ' f i ' + 1 6 7 ' R ' + 16P=R= + 3907=7' + l+4S'7= - 407=«» + I 6 7 ' s ' -mp'R' - I60P'S= + l6P=fi + I 6 S ' P + 14Pfl' - 407»R=) , o (34)

Fram 14, Ch. 34. pp. 189-190). we note dial for o = I/25 and using die fact that G. = t , , , . we have Gj = I and G25 = C1/2J = ^ | t l .

Employing die values of G, and C , , : we can compute the values of 7 and R, and then einp^oymg the resulting values of 7 and R in (34). we find that die third factor vanishes for a .specific value of, = e - ' / ' , dius by die identity theorem it vanishes identically Q Theorem 3lfa./l.y. and S have degrees 1.5.5. and 25. respectively then

(7=.+ ^ ) . | 4 ( 7 . ' + J , ) - 2 9 ( 7 . = + ^ ) - l 4 4 ( 7 . + ± ) -259(7* + J , ) - , 6 ( p ' + J , ) [ ( 7 . + ^ ) + (7* + ± ) + | ] + « ( ' V ± ) [ ( 7 . = + J , ) + 2 ( 7 » + - L ) „ ( 7 * + ± ) + 4 ] = 390. ,35,

S : l wrbS^I)' "" " '" '"'• ' """"•' ""'"""' »""=" " ^^»^" •" '/'•'

a

S Spnnger

510 1 M. S. Mahadeva Naika et al, ^

Theorem 4 If a. fl, y.ands have degrees I 5,4. and 20, respectively, then

( C * + ^ ) - 56 (2-= + J ^ ) + 861 ( 2 - + ±,) - 5824 ( e " + ± ) +22524 ( e ' + ^ ) - 59015 (2* + J j ) + ,02884 (2= + J j ) - 127634

^<'^'^?)-(<^'-^)]-(^'-^)h(fl= + ±)

- 6 2 8 ( 2 * + ± ) + 2 2 4 ( 2 ' + i , ) - 2 4 ( 2 . + J , ) _ „ 9 6 ] + (r= + ± ) [2228 (fl + 1 ) - ,800 (e= + J , ) + ,26, (e= + J , ) - 3 3 6 ( 2 ' + ^ ) + , „ , ( 2 ' + J , ) - , 6 ( 2 . . + ^ ) ] .

P ™ / Replacing , by ,* in (20), m and a changes to „,• and 4. respectively dien we anive ,»' + 4=£!^ + |,

wherem' = J&J-and6 := |i!i=a I

a-iatO, \nt-y)\

s a m l T u f " f h ™ ' " , ' ?^ ' " * = " " • " ° ° ' « " " <3«' S""^= »•= P'-of °f (36) runs on the

same line as the proof of Theorem 1, so wc omit die details p Theorem 5 Ifot, H. y. ands have degrees I, 5. 4, andlO, respectively, then

7== - 30727=* - 4445447" - ('70328327= + ^"38688 N rf(«VJ,)[2'(l96,P* + ^ ) , 3 4 ( 5 9 5 P ' = + f ^ ) + 3 5 5 7 = . ]

- • ( " ' - F ) [ 0 - 3 P = + ^ ) + 8(„,p,',£li),3,3,J

- ' ( « " . - i , ) [ ( l 0 3 3 p l = + i : ^ ) , , , , ( 3 2 P * + - ) j

- " ( . • ' + 1 ) [ 3 ( | 3 P ' + - ) + , 9 0 ] + 2 I * ( R = . + J ^ )

[ " ^ ' ^ ^ ] ^ ^ " ( ^ ^ ' - ^ ) - ^ ( « ^ ' - ^ ) = 3 3 5 5 4 4 3 2 . ,3,,

Proof nie proof of (38, is similar to the proof of (27,. Hence, we omit the details. •

© Springer

in New Sdilafli Type Mixed Modular Equation

Theorem 6 If a, p. y, and S have degrees 1.5.4. and 20. respectively, then 26738688 \ P^^ - i012P^* - 444544P'^ - f 7032832/"^ -f- -

+ 2 ' ( r * + ± ) [ 2 ' ( l , 6 1 P * + ^ ) + 24 ( 5 , 5 P " + f l » ) +355P=»]

- 2 - ( 7 . + i , ) [ ( l 6 4 3 3 7 ' + 5 f ^ ) + 8 ( l l l p . ' + i l i ) + 8 4 3 5 2 ] + 2 ' ( 7 . = + ^ ) [ ( l „ 3 3 P ' = + H g ) + 2 2 4 ( 5 2 P * + - ) ] - 2 . * ( 7 » + l ) [ 3 ( l 3 7 ' + ^ ) + 2 9 0 ] + 2 " ( 7 = " + ^ )

[257* + * ] - 2 " (7=* + ^ ) + ^ ( 7 = ' + ^ ) = 335544.32. (39) Proof Interchanging /( and >- in (38), P remains unchanged whereas R changes to l / T .

Hence, we obtain (39). Q Theorem 7 If a. fi. y. and S luive degrees I. 5. 11, and 55. respectively, then

[7' + J,] + ll(3[7 + l]+8[7= + J,] + [7= + J,]-[7* + J , l - l 5 |

Proof Replacing ^ by ^ ' ' in equaUon (20), m and a changes to m' and b. respectively, then weamve

where/n':= i ^ and fc : - I - ^ ! | ^ r

Employing (20) and (41) in (13). we obtain (40) Since the proof of (40) nins on the same

nne as the proof ofTheorem I, so we omit the details. • Theorem 8 If a. fi. y. andS have degrees I. 5. II. and 55. respectively, then

( « " - ^ ^ ) + n6(R=»+^)+3872(R"+J,)+25476(R'=+Ji3) + I 8 I 9 8 4 ( R ' + ± ^ + 4 7 2 5 6 O ( R * + J ^ ' ) + I 2 9 5 5 5 8 - 1 0 2 4 / ' P = » + - L ] + ( ' • " + 7 ^ ) [»32 (R* +J,)+25344]-(p.= + - L ) [53504 (R* + ± ) +10560(R»+J^j + 138688] + (p«+-^)[217536(R*+i;j+45584 X ( R « + J J ) + T O 6 ( R " + J ^ ) + 4 3 0 4 9 6 ] - ( 7 * + J J ) [406736 ( R * + ± ) + 1 2 6 2 8 0 ( R - + J , ) + 1 9 S 8 8 ( R ' = + J ^ ) + 2 3 7 6 ( P « + J ^ ) + 9 6 2 0 I 6 ] = 0 . ( 4 2 )

Q Springer

~ M. S Mahadeia Naika et at ^ Proof The proof of (42) is similar to the proof of (27). Hence, we omit die details. n

Theorem? Ifa.ti. y. and Shave degrees 1, 5, U. and 55. respectively, then (!"=* + j L ) + 176 (7=» + ^ ) + 3872 ( 7 " + ^ ) - ^ 25476 ( 7 " + ^ ) +181984 (7» + J , ) + 472560 ( 7 * + J , ) + 1295558 - 1024 (?=» + - i , \ + ( ' • " + 7 ^ ) [5632 ( 7 * + ^ ) + 2 5 3 4 4 ] - ( ? • = + J , ) [53504 ( 7 * + J , ) +10560 (7= + J , ) + 138688] + (p» + J , ) [217536 ( T * + 5L) + 45584

" {^' + Tt)+™f>{l"^Yn) +430496] - ( P * + J , ) [406736 ( 7 * + - L ) + 126280 (7» + J , ) + 19888 (71= + a , ) + 2 3 7 6 ( 7 " + J , ) + 962016] l o 7roo/ Interehangiug ^ and y in (42). 7 remains unchanged whereas R changes to 1/7

Hence, we obtain (9). ' " ' / J ,

»r^vejr;x"ss:r?^:t:::s;^ssrs:;rssrt:

Technelog, (DST), New D.II,,, M i . fo, si,pp„„ . , < , „ fc , „ „ „ i p „ j „ s R / s S " , !

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' m S S ( * o " " ° ' " ' " " " = " ' • S=M.li'-.>l.e - m . x . T ™ d a l „ . , . . , i o . s . , Number T l , „ „ I « " ? • S ' J ' " " " " ' • ' • ^ Noiebooks. P . „ III. Sp„„Ber, New York (1991)

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ScLiii Comemp. Matli 21 189-206(2011) "^ "^* ^'^""'f"-'yP= mixed modular equations Adv.

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0 Springer