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Bivariate Series of Power-Trigonometric Type

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Bivariate Series of Power-Trigonometric Type Bivariate Series of Power-Trigonometric Type Mohammad Masjed-Jamei a Wolfram Koepf b,* a Department of Mathematics, K.N.Toosi University of Technology, P.O.Box 16315-1618, Tehran, Iran, E-mail: [email protected] , [email protected] b,* Department of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany, E-mail: [email protected] , Corresponding author * Abstract. Let k k Cr (, ) ark cos( k ) and Sr (, ) ark sin( k ) k 0 k 0 be two bivariate series in which r, are real variables, and {}akk0 are known coefficients. In this paper, we first obtain the explicit forms of Cr (, ) and Sr (, ) for ()k afakk ()/! where f is an infinitely differentiable function at x a and then obtain three new integral formulas in this sense. In the sequel, if {}akk0 are of hypergeometric type, Cr (, ) and Sr (, ) will be reduced to two trigonometric- hypergeometric series that can generate various kinds of new trigonometric series. By defining these two specific series in the last section, we obtain some of their important properties such as integral representations and several classes of partial differential equations. The convergence of each introduced series is also studied. Keywords. Bivariate series of power-trigonometric type, Fourier trigonometric series, hypergeometric series, Convergence radius, Dominated convergence theorem. Mathematics Subject Classification: 33C20, 33C05, 42A16, 42A32, 42B05, 65B10, 65D20, 40A30. 1. Introduction It is well-known that there are some differences between power series expansion and Fourier trigonometric expansion. For example, the infinite differentiability of a function does not itself assure that its power series will converge to that function, whereas mere periodicity and a little smoothness are enough to have the Fourier trigonometric series converge uniformly to the function [3]. Further, the terms of the trigonometric series expansion describe simple harmonic motion so that the function may be considered as a linear combination of harmonic motions, whereas the terms of a power series have no such physical interpretation. Beyond the fact that both Taylor and Fourier series are infinite, it is pointed out in [3] that there are some similarities between these two series too. Indeed, an expansion of a real function in terms of a series of ultraspherical polynomials [12] is found in [3] that contains only cosine type of Fourier series and power series as special limiting cases. 1 In this paper, we follow a new approach to introduce a mixed type of power and trigonometric series, which extends both Taylor and Fourier series expansions simultaneously. We then focus on a special case to define two trigonometric types of hypergeometric series that cover many ordinary hypergeometric series together with Fourier trigonometric series. Some illustrative examples are given in this sense and three new integral formulas are derived. In the sequel, we obtain some important properties of the two introduced series such as integral representations and several classes of partial differential equations. Let us start our treatment with defining two bivariate series as k k Cr (, ) ark cos( k ) and Sr (, ) ark sin( k ) , k 0 k 0 in which , {}akk0 are known numbers and r, are real variables. cos( k 1) sin( k 1) By noting that lim and lim are not known, according to k cos( k ) k sin( k ) Abel’s theorem [5], the convergence of Cr (, ) and Sr (, ) would directly depend on k the convergence radius of ark because k0 cos( k ) 1(k and ). sin( k ) k This means that if ark is convergent in a specific region, Cr (, ) and Sr (, ) are k0 also convergent in that region for any [,], which is the standard orthogonality interval of the Fourier trigonometric expansion. f ()k ()a Now, assume that f is a successively differentiable function at x a . For a k k ! we can obtain the explicit values of the two series Cr (, ) and Sr (, ) in terms of the given function f . For this purpose, let zxiye i (1) i be a complex y number in which xy22, arctan and xy, and then suppose that x f :2 can be expanded as follows k f ()zcz k , (1) k 0 where {}ckk0 are all pre-determined numbers. Since zzkk kikikk()2cos e e k, and zzkk 2sin k k , i it is directly concluded from (1) that zfz() zfz () Imzfz ( ) zfz ( ) Im( ) 0, . (2) i 2 In other words, zfz() zfz () and (()())/zfz zfz i and any linear combination of them are always real functions if (1) holds. By noting (2), let us now define a very general bivariate real function as A Hr(,;,;,) AB eiiii fa ( re ) e fa ( re ) 2 (3) B egbreeiiii() gbre ( ), 2i where f , g are analytic functions at respectively ra and rb , , , A, B are known constants and r, are real variables. Since k dHr(,;,;,) AB ()kk () k Af()cos( a k ) Bg ()sin( b k ), dr r0 the Taylor expansion of the function (3) with respect to the variable r at r 0 is as AB eiiii f() a re e f ( a re ) e iiii g () b re e g ( b re ) 22i (4) r k Af()kk()cos( a k ) Bg () ()sin( b k ) . k 0 k! Clearly (4) generates the same as Taylor expansion for 0 . Also, in terms of the ** variable , it is reduced to a Fourier series for 0 as akbkkkcos sin k 0 r k r k where aAfa*() k () and bBgb*() k () . For B 0 and A 0 , formula (4) is k k! k k! finally reduced to the following forms 1 r k efareeiiii() fare ( ) f() k ()cos() a k , (5) 2!k 0 k and 1 r k efareeiiii() fare ( ) f() k ()sin() a k . (6) 2!ikk 0 As was pointed out, relation (4) is valid only in the convergence region of r for any , and [,]. Furthermore the initial conditions k k *()k r *()k r limaAk lim fa ( ) 0 and limbBk lim gb ( ) 0 , kk k! kk k! must be simultaneously satisfied. In this case, we automatically have * * limakk cos( ) 0 and limbkk sin( ) 0 . (7) k k 3 * By accepting the conditions (7), the convergence of two series akk cos( ) and k 0 * bkk sin( ) would directly depend on the convergence radius of the Taylor k 0 expansions corresponding to f and g at x a and x b . Example 1. If f (ze ) z and gz( ) ln( z 1) at ab 0 , then we have 1 efreeii( ) i fre ( i ) e rcos cos( r sin ), 2 (8) 1 efreeii( ) i fre ( i ) e rcos sin( r sin ), 2i and 1cossinr eii g( re ) e i g ( re i ) ln( r2 1 2 r cos ) sin arctan ( ), 22 1cos r 1sinsinr eii g( re ) e i g ( re i ) ln( r2 1 2 r cos ) cos arctan ( ). 22ir 1cos (9) Now, choosing AB 1 in (4) gives r k cos(kk ) ( 1)k ( 1)! sin( k ) k! k 0 (10) sinr sin err cos cos( sin ) ln( rr 2 1 2 cos ) cos arctan ( ), 21cos r r k which is a particular case of (4). On the other hand, since lim 0 for any r and k k! r k lim( 1)k 1 0 for r [ 1,1], relation (10) is valid only for r [ 1,1] [ 1,1] , k k [,] and , . Another example, derived from (8) and (9), may be considered as r k sin(kk ) ( 1)k ( 1)!cos( k ) k! k 0 cosr sin err cos sin( sin ) ln( rr 2 1 2 cos ) sin arctan ( ), 21cos r which is again valid for r [1,1], [,] and , . 1.1. Three new integral formulae derived by the expansion (4) It is in general known that {cos( k ) }k0 and {sin( k ) }k1 constitute two biorthogonal sequences [5,8] for 0 and 1/ 2 on [ , ] so that we have 4 0(j k ), cosjkd cos 2 cos jkd cos 0 jk, (),j k sinjkd sin 2 sin jkd sin , 0 jk, sinjkd cos 0, jk , , and 11 11 cos(jkdjkd ) cos( ) 2 cos( ) cos( ) j,k , 220 22 11 11 sin(jkdjkd ) sin( ) 2 sin( ) sin( )jk, , (11) 220 22 11 sin(jkdjk ) cos( ) 0, , , 22 where [ , ] can be changed to any other arbitrary interval by a simple linear transform. By referring to expansion (4), now define the finite sequence l j ()jj () r TAfajBgbjl () ()cos( ) ()sin( ) , j0 j! and assume that there exists a real function t( ) ( independent of l ) such that Ttl () () . Since efareeiiii() fare ( ) and egbreeiiii() gbre ( ), are respectively even and odd functions in , using the dominated convergence theorem [5] would yield the following integral formulae 1 efareeiiii() fare ( )cos kd 2 jj ()jjrr () f ()ajkdfajkd cos( ) cos () cos( ) cos jj00jj!! r k fa()k () ( k {0}), ()!k (12) j 1 r f ()()areii fare d f() j ()cos a j d 00 2!j0 j (13) j r fa()j () cos jd fa (), 0 j0 j! and 5 1 egbreeiiii() gbre ( )sin kd 2i jj ()jjrr () gb( )sin( j ) sin kd gb ( ) sin( j ) sin kd jj00jj!! r k gb()k () ( k {0}). ()!k (14) Similarly, corresponding to relations (11) we obtain 11 efareeiiii/2() /2 fare ( )cos() k d 22 j 11r (fa()j ( )cos( j ) ) cos( k ) d (15) j0 2!j 2 jk rr 11 fa()jk() cos( j ) cos( k ) d fa () () ( k {0}), j0 jk!22 ! and 11 egbreeiiii/2() /2 gbre ( )sin() k d 22i j 11r (gb()j ( )sin( j ) ) sin( k ) d (16) j0 2!j 2 jk rr 11 gb()jk() sin( j ) sin( k ) d gb () () ( k {0}). j0 jk!22 ! Corollary 1. If m {0} and n {1 / 2} except the case mn 0 , then three following integral formulae would be totally derived from relations (12) – (16): m 1 r efareeni() i ni fare ( i )cos() mnd f() m (), a 0 22!m 1 fa()() reii fa re d fa (), 0 2 m 1 r efareeni() i ni fare ( i )sin() mnd f() m ().
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