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Bivariate 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 {}akk0 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 formulas in this sense. In the sequel, if {}akk0 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 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 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    , {}akk0 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 k0 cos( k ) 1(k  and  ). sin( k )  k This means that if ark is convergent in a specific region, Cr (, ) and Sr (, ) are k0 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 {}ckk0 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 () Imzfz ( ) 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 eiiii fa ( re  ) e  fa ( re  ) 2  (3) B egbreeiiii()   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 r0 the Taylor expansion of the function (3) with respect to the variable r at r  0 is as

AB eiiii 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 efareeiiii()   fare (   )  f() k ()cos() a  k , (5) 2!k 0 k and 1  r k efareeiiii()   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 efreeii(  )  i fre ( i  ) e rcos  cos( r sin ), 2  (8) 1 efreeii(  )  i fre ( i  ) e rcos  sin( r sin ), 2i  and 1cossinr eii g( re  ) e  i g ( re i  ) ln( r2 1 2 r cos ) sin arctan ( ), 22 1cos r 

1sinsinr  eii 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) sinr 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 cosr 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 ) }k0 and {sin( k ) }k1 constitute two biorthogonal [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

l j ()jj () r TAfajBgbjl () ()cos( ) ()sin( ) , j0 j! and assume that there exists a real function t ( ) ( independent of l ) such that

Ttl ()  () . Since

efareeiiii() fare (  ) and egbreeiiii() gbre (  ), are respectively even and odd functions in  , using the dominated convergence theorem [5] would yield the following integral formulae

 1 efareeiiii()   fare (   )cos kd  2 jj   ()jjrr () f ()ajkdfajkd cos( ) cos () cos(  ) cos     jj00jj!! r k  fa()k  () ( k  {0}), ()!k  (12) j  1 ii () jr f ()()are fare d  f ()cos a j d 002! j j0 (13) j  r  fa()j () cos jd  fa (),  0  j0 j! and

5  1 egbreeiiii()   gbre (   )sin kd  2i jj   ()jjrr () gb( )sin( j ) sin kd gb ( ) sin(  j ) sin kd      jj00jj!! 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)   j0 2!j 2 jk  rr 11 fa()jk() cos( j ) cos( k ) d fa () () ( k {0}),   j0 jk!22 ! and  11 egbreeiiii/2() /2 gbre (  )sin() k  d  22i j   11r (gb()j ( )sin( j ) ) sin( k ) d (16)   j0 2!j 2 jk  rr 11 gb()jk() sin( j ) sin( k ) d gb () () ( k {0}).   j0 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 (). a 0 22!im m1 Example 2. Let f ()zz 1 with fj()mm(0) ( 1) (  1/ 2). We have j0 11sin1 r  efreeni( i ) ni  fre ( i  ) ( r2 1 2 r cos )4 cos( n arctan ( )) , 221cos r  and 11sin1 r  eni f() re i e ni  f ( re i  )(12cos)sin(arctan( r2 r4 n )). 221cosir 

Therefore, according to corollary 1, three

6 1 m  1sinrr () m1 (rr2  1 2 cos )4 cos( n  arctan ( ))cos( mnd  )   ( j  1/ 2), 0  21cos rm 2!j0 1  1sinr  (rr2  1 2 cos )4 cos( arctan ( )) d  , 0 21cos r  1 m  1sinrr () m1 (rr2  1 2 cos )4 sin( n  arctan ( ))sin( mnd  )   (1/2),j  0  21cos r  2m! j0 are valid for any r [1,1] ,  [0, ] , n {1 / 2} and m {0} except the case 1 mn0 . Notice that (.) 1. j0

2. Power-Trigonometric Series of Hypergeometric type

It is well known that the generalized hypergeometric series (as special cases of a power series) are fundamental tools for investigations in different branches of engineering and mathematical sciences, see e.g. [1, 2, 11, 13]. For instance, there is a large set of hypergeometic-type polynomials whose variable is located in one or more of the parameters of the corresponding hypergeometric functions [6]. These polynomials are of importance in mathematics as well as in some areas of physics. A few examples of their applications are discussed by Nikiforov, Suslov and Uvarov [10]. See also [7, 9, 12]. Essentially, the main motivation for introducing and developing hypergeometric series is that many elementary and familiar functions (such as trigonometric, exponential and logarithm functions, classical orthogonal polynomials of Jacobi, Laguerre and Hermite, etc. [9, 12]) can be written in terms of them and therefore their initial properties can be found via the initial properties of hypergeometric functions. They also appear as solutions of many important ordinary differential equations [2, 11]. The generalized [2]

 k aa12, , ... ap (aa ) ( ) ...( a ) z Fz 12kkpk, (17) pqbb, , ... b  12 q k0 (bb12 )kkqk ( ) ...( b ) k ! k 1 in which ()rrjrkrk  ( ) ( )/() and z may be a complex variable is indeed a j0  k ()k expansion for a function, say f, as k z with k  f (0) /k ! for k0 which the ratio of successive terms can be written as

 (kaka )( )...( ka ) k1  12 p . kq(kbkb 12 )( )...( kbk )( 1)

According to the well-known ratio test [5], series (17) is convergent for any p q 1. In other words, it converges in z 1 for p  q 1, converges everywhere for p q 1

7 and converges nowhere ( z  0 ) for p q 1. Moreover, for p  q 1 it absolutely converges for z 1 if the condition

qq1 *  AbaRejj 0 , jj11 holds and is conditionally convergent for z 1 and z  1 if 10A* and is finally divergent for z 1 and z  1 if A*  1. One of the most important cases of (17) arising in many physical problems [9, 13] is the Gauss hypergeometric series, which is convergent in z 1 and is defined by

k ab,  ()()abzc() 1 Fzkk tttzdtcbbcba11(1  ) (1 ) (Re Re 0), 21 0 c k 0 ()ckk ! () bcb ( ) (18) satisfying the differential equation

zz1(1) y a b z cy  aby 0. (19)

In this section, by referring to the basic relations (5) and (6), we extend the hypergeometric series (17) and respectively define a special case of the series (5) as

 k aa12, , ... ap (aa ) ( ) ...( a ) r Cr(, );12kkpk cos( k ), (20) pqbb, , ... b  12 q k0 (bb12 )kkqk ( ) ...( b ) k ! and a special case of the series (6) as

 k aa12, , ... ap (aa ) ( ) ...( a ) r Sr(, );12kkpk sin( k ). (21) pqbb, , ... b  12 q k0 (bb12 )kkqk ( ) ...( b ) k !

According to the explained themes, both series (20) and (21) are convergent if the corresponding series (17) is convergent. It is not difficult to verify that

aa12, , ... app aa 12 , , ... a CrCr(, ); cos  (,  );0 pqbb, , ... b pq bb , , ... b 12qq 12 (22) aa12, , ... ap  sinSr ( ,  );0 , pqbb, , ... b 12 q and

aa12, , ... app aa 12 , , ... a SrCr(, ); sin  (,  );0 pqbb, , ... b p q bb , , ... b 12qq 12 (23) aa12, , ... ap  cosSr ( ,  );0 . pqbb, , ... b 12 q

8 The main motivation for defining such bivariate series may be this fact that many ordinary hypergeometric series (when   0 ) and Fourier trigonometric series (when r is fixed and pre-assigned) are representable in terms of the series (20) or (21).

Example 3. The fact that a discontinuous function can be represented as an infinite sum of sines and cosines was known already to Euler [4], who obtained the following sum

  sin k   ,0 2 . (24) 2 k 1 k

But (24) can be now represented as

  sinkj sin( 1)(1)jj (1) sin( j 1) 1, 1   21S (1, );1 . kj10kj1(2)! j  0j j2

Example 4. Since 1, 1 ln(1xiy ) 21Fxiy 2 xiy xyyy (25) ln (1x )22yy arctan x arctan  ln (1 xy )22 2112xxi , xy22 xy 22 replacing xr cos and yr sin in (25) gives

1cos2 r sin 1, 1 ln 1rr 2 cos sin arctan ( ) 21 C ( r ,  );0 , rr21cos  2 and

1sinsinr 2 1, 1 cos arctan ( ) ln 1rr 2 cos21 S ( r , );0 . rr1cos  2 2

Therefore, by noting (22) and (23), we can respectively obtain

1, 1  r k 21Cr(, ); cos(  k )  2  k 1 k 0 sin( 1)r sin  cos(  1) arctan ( )  ln 1 rr2  2 cos , rrr1cos  2 and 1, 1  r k 21Sr(, ); sin(  k )  2  k 1 k 0 (26) cos( 1)r sin  sin(  1) arctan ( ) ln 1rr2 2 cos . rrr1cos  2

9 It is interesting to note that substituting r   1 in the right hand side of (26) exactly gives the left hand side of (24).

Example 5. Since

1/2, 1 2 arctan(xiy ) 21Fxiy()  3/2 xiy 1(1)2yx22 y x (lnx arctan ) 2(xy22 ) 2 x 2 (1 y ) 2 1 xy 22 1(1)2xx22 y x iy(ln arctan ), 2(x22yxy ) 2 2 (1 ) 2 1 xy 22 so  (1)k 1 sin 1rr2 2sin 2cos r rk2k cos 2 ( ln cos arctan ( ))  21krrrr 2 2 122 2sin 1 k 0 1/2, 1 2 21Cr(,2);0, 3/2 and  (1)k 1 cos 1rr2 2sin 2cos r rk2k sin 2 ( ln sin arctan ( ))  21krrrr 2 2 122 2sin 1 k 0 1/2, 1 2 21Sr(,2);0. 3/2

By noting (18), we can obtain the integral representation of series (20) and (21) for (,)pq  (2,1) respectively as follows: Since

1sina /2 tr  (1tr eia ) (1 tr e ia ) 1 t22 r 2cos tr cos(arctan a ) , 2cos1tr   so  ()()abr k kk cos( k )  ()ck ! k0 k ()ctr1 a/2 sin ttbcb1122(1 ) 1 trtr 2 cos cos( a arctan ) dt . ()(bcb )0 tr cos  1

Similarly, by noting (18), the integral representation of series (21) for (,)pq  (2,1) takes the form  ()()abr k kk sin( k )  ()ck ! k 0 k ()ctr1 a /2 sin ttbcb1122(1 ) 1 trtr 2 cos sin( a arctan ) dt . ()(bcb )0 tr cos  1

10 2.1. Partial differential equations of series (20) and (21) There exist several classes of partial differential equations (PDEs) for each bivariate series (20) and (21). We first obtain the equations for (,)pq  (2,1) and then extend the results to the general case (,)pq . Let us begin with the assumption

ab, 21Cr(, ); yry (,  ) . (27) c

The following equations hold for y defined in (27):   ()ab () r k ()()rarby kk11 cos()  k , (28) rrk 0 () ckk !   ()ab () r k ()()()() ar  by  br  ay  kk11 sin(),  k rrckk 0 ()k ! and (29)  ()()abrrkk1 () a () b (1)rcy kk cos()  k  k11 k cos(1)  k rr() ck (  1)! () ck ! kk10kk1 ()ab ()rrkk () ab () coskk11 cos()sinkk kk  11 sin(). kk00()ckkk ! () ck ! (30) By substituting (27) and (28) into (29) one respectively obtains

     (1)(cos()()sin()())rcy rarb       arb  y , (31) rr  r  r   r and      (1)(cos()()sin()())rcy  rarb       bray  . (32) rr  r  r   r

Another approach to obtain further classes of PDEs is to switch with the variable  such that we have    ()ab () r k ()()()() ab  y  ba  y  kk11 cos(),  k    k 0 ()ckk ! (33)    ()ab () r k ()()()()ra  byrb    ay  kk11 sin(),  k rr  k 0 () ckk ! and (34)   ()()ab r k ()(1)cy kk cos()   k  k 1 ()ckk 1 (  1)!  ()ab () r k 1  kk11 cos(k 1) (35) k 0 ()ckk ! ()ab ()rrkk () ab () rkrkcos kk11 cos()sin  kk  11 sin().  kk00()ckkk ! () ck !

11 Again, substituting (33) and (34) into (35) yields the following equations

    ()(1)(cos()()sin()()),  cyr    a    br   r  a    by      r  (36)     ()(1)(cos()()sin()()),  cyr    a    br   r  b    ay      r  (37)     ()(1)(cos()()sin()()),  cyr    b    ar   r  a    by      r  (38)     ()(1)(cos()()sin()()).cyr   b  ar  r  b  ay      r  (39) Finally a third approach may be a simultaneous combination of relation (28) with (33) and (29) with (34) so that we have

     (1)(cos()()sin()()),rcy    a     b       arby  rr     r (40)      (1)(cos()()sin()()),rcy   ab           bra  y rr      r (41)      (1)(cos()()sin()()),rcy   b    a      arby  rr     r (42)      (1)(cos()()sin()()).rcy  ba         bra  y rr      r (43) Corollary 2. After simplifying and computing each ten equations in (31), (32) and (36) – (43), the explicit forms of PDEs for series (27) are respectively as follows:

22yy y rr(cos ) 12 r sin  ( ab 1)cos  ( arc )sin rr  r (31a) y bbaaysin ( cos ( )sin ) 0,  22yy y rr(cos ) 12 r sin  ( ab 1)cos  ( brc )sin rr  r (32a) y aabbysin ( cos ( )sin ) 0,  22yy  y rrcos 122 sin  ( br ) sin 2 rr y ()cossin1ba  a rc  (  ba )(sin()cos)(1)0,   a r  c y  (36a)

12 22yy  y rrcos 122 sin  ( ar ) sin 2 rr y ()cossin1ba  b rc  (  ab )(sin()cos)(1)0,   b r  c y  (37a) 22yy  y rrcos 122 sin  ( br ) sin 2 rr y ()cossin1ba  a rc   (   ba )(sin()cos)(1)0,    a r  c  y   (38a) 22yy  y rrcos 122 sin  ( ar ) sin 2 rr y ()cossin1ba  b rc   (   ab )(sin()cos)(1)0.    b r  c  y   (39a) 22yy  2 y  y rrarc22cos sin  (  ) sin rr   r (40a) y (ba )cos  b sin  ( a  ) ( b  )cos  b sin y  0,  22yy  2 y  y rrbrc22cos sin  (  ) sin rr   r (41a) y ()cossin(ba  a  b  )()cossin0, a   a y   22yy  2 y  y rrarc22cos sin  (  ) sin rr   r (42a) y (ba  )cos  b sin  ( a   ) ( b  )cos  b sin y  0,  22yy  2 y  y rrbrc22cos sin  (  ) sin rr   r (43a) y (ba  )cos  a sin  ( b   ) ( a  )cos  a sin y  0. 

It is interesting to note that replacing   0 in (31a) and (32a) exactly gives the same as Gauss differential equation (19).

The corollary 2 shows that there are various classes of PDEs for the general case (,)pq  (2,1). Due to pages limitation, we here obtain only one class. This turn assume that

aa12, , ... ap Cryry(, ); (,  ) . (44) pqbb, , ... b 12 q

The following equations hold for y defined in (44):

13  k   (aa11 )kpk ...( ) 1r (rararay12 )( )...( p ) cos(  k ) , rr  rk 0 ( bbk1 )kqk ...( ) !   (ara12 )(  )...( r  ayp )  ( ar 213 )(  ara )(  )...( r  ayp ) rr  rrr  k  (aa11 )kpk ...( ) 1r ... (  arpp )(  ar12 )(  a )...( r  a 1 ) y  sin(  k ) .  rr rk 0 ( bbk1 )kqk ...( ) ! Consequently we have  k1    (aa1 )kpk ...( ) r (rbrb12 1)(  1)...( rb q 1) y  cos(  k ) rr  r  rk 1 ( b11 )kqk ...( bk ) 1 (  1)!  (aa ) ...( ) r k  11kpk 1 cos(k 1) k 0 (bbk1 )kqk ...( ) !    cos (rararay12 )( )...(p ) sin ( ar12)(  a )...( r  ap ) y rr  r  rr      cos (rararay12 )( )...(p ) sin ( arararay 213 )( )( )...(p ) rr  r   rr  r      cos (rararay12 )( )...(pp ) sin ( ar )( ar12)( a )...( r ap 1 ) y . rr  r   rrr Similarly, if we take

aa12, , ... ap Sryry(, ); (,  ) , (45) pqbb, , ... b 12 q then we obtain  k   (aa11 )kpk ...( ) 1r (rararay12 )( )...( p ) sin(  k ) , (46) rr  rk 0 ( bbk1 )kqk ...( ) !

  (ar12 )(  a )...( r  ap ) y  (  ar 213 )(  ar )(  a )...( r  ap ) y rr rrr  k  (aa11 )kpk ...( ) 1r ... (  arpp )(  ar12 )(  a )...( r  a 1 ) y  cos(  k ) .  rr rk 0 ( bbk1 )kqk ...( ) ! Therefore (47)  k 1    (aa1 )kpk ...( ) r (rbrb12 1)(  1)...( rb q  1) y  sin(  k ) rr  r  rk 1 ( b11 )kqk ...( bk ) 1 (  1)!  (aa ) ...( ) r k  11kpk 1 sin(k 1) k 0 (bbk1 )kqk ...( ) !

   cos (rararay12 )( )...(p ) sin (  ar12)(  a )...( r  ap ) y rr  r  rr      cos (rararay12 )( )...(p ) sin ( arararay 213 )( )( )...(p ) rr  r   rr  r      cos (rararay12 )( )...(pp ) sin ( ara )(1 )(raray21 )...(p ) . rr  r   rrr (48)

14 Corollary 3. Let (,)pq  (2,1) in (45). After applying the relations (46) – (48), ten ab, different PDEs would appear for 21Sr(, );  y as follows: c 22yy  y rr(cos ) 12 r sin  ( aba 1)cos  ( )sin rc rr  r y bbaaysin ( cos ( )sin ) 0,  22yy  y rr(cos ) 12 r sin  ( abb 1)cos  ( )sin rc rr  r y aabbysin ( cos ( )sin ) 0,  22yy  y rrbrcos 122 sin  ( ) sin 2 rr y (ba )cos  a sin rc  1 ( b  )(( a  )cos  a sin ) r  (  1 cy ) 0,  22yy  y rrarcos 122 sin  ( ) sin 2 rr y (ba )cos  b sin rc  1 ( a  )(( b  )cos  b sin ) r  (  1 cy ) 0,  22yy  y rrbrcos 122 sin  ( ) sin 2 rr y (ab )cos  a sin rc  1 ( b  )(( a  )cos  a sin ) r  ( 1 cy ) 0,  22yy  y rrarcos 122 sin  ( ) sin 2 rr y (ab )cos  b sin rc  1 ( a  )(( b  )cos  b sin ) r  ( 1 cy ) 0, 

22yy  2 y  y rrbrc22cos sin  (  ) sin rr   r y ()cossin(ba  a  b  )()cossin0, a   a y   22yy  2 y  y rrarc22cos sin  (  ) sin rr   r y (ba )cos  b sin  ( a  ) ( b  )cos  b sin y  0,  22yy  2 y  y rrbrc22cos sin  (  ) sin rr   r y ()cossin(ab  a  b  )()cossin0, a   a y  

15 22yy  2 y  y rrarc22cos sin  (  ) sin rr   r y (ab )cos  b sin  ( a  ) ( b  )cos  b sin y  0. 

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