Computers and Mathematics with Applications 56 (2008) 2941–2947 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa A generalization of Fourier trigonometric series Mohammad Masjed-Jamei a,b,∗, Mehdi Dehghan a a Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No 429, Hafez Ave, Tehran, Iran b Department of Applied Mathematics, K.N.Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran article info a b s t r a c t Article history: In this paper, by using the extended Sturm–Liouville theorem for symmetric functions, we Received 30 August 2007 introduce the differential equation Received in revised form 31 May 2008 C Accepted 10 July 2008 00 n 2 a.a 1/ n Φ .t/ C n C a.1 − .−1/ /=2 − .1 − .−1/ /=2 Φn.t/ D 0; n cos2 t Keywords: f g1 Extended Sturm–Liouville theorem for as a generalization of the differential equation of trigonometric sequences sin nt nD1 and f g1 D symmetric functions cos nt nD0 for a 0 and obtain its explicit solution in a simple trigonometric form. Symmetric orthogonal functions We then prove that the obtained sequence of solutions is orthogonal with respect to the Norm square value constant weight function on T0; πU and compute its norm square value explicitly. One of the Fourier trigonometric sequences important advantages of this generalization is to find some new infinite series. A practical Hypergeometric functions example is given in this sense. ' 2008 Elsevier Ltd. All rights reserved. 1. Introduction Many sequences of special functions are solutions of a usual Sturm–Liouville problem [1,2]. For instance, the well-known f g1 f g1 trigonometric sequences sin nx nD1 and cos nx nD0, which respectively generate the Fourier sine and cosine series, are solutions of a usual Sturm–Liouville equation in the form 00 C 2 D 2 T U Φn .x/ n Φn.x/ 0; x 0; π : (1) It is clear that the interval T0; πU in equation (1) can be transformed to any other arbitrary interval, say [−l; lU with period 2l, by a simple linear transformation. On the other hand, most special functions applied in physics, mathematics and engineering, satisfy a symmetry relation as n C Φn.−x/ D .−1/ Φn.x/ 8n 2 Z : (2) Recently in [3], we have presented a key theorem by which one could generalize the usual Sturm–Liouville problems with symmetric solutions. We have shown that the solutions corresponding to an extended Sturm–Liouville equation are orthogonal [4,5] with respect to an even weight function on a symmetric interval. In other words: n Theorem 1.1 ([3]). Let Φn.x/ D .−1/ Φn.−x/ be a sequence of symmetric functions that satisfies a differential equation of the form 00 C 0 C C C − − n D A.x/Φn .x/ B.x/Φn.x/ λnC.x/ D.x/ .1 . 1/ /E.x/=2 Φn.x/ 0; (3) ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (M. Masjed-Jamei), [email protected] (M. Dehghan). 0898-1221/$ – see front matter ' 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2008.07.023 2942 M. Masjed-Jamei, M. Dehghan / Computers and Mathematics with Applications 56 (2008) 2941–2947 where A.x/; B.x/; C.x/; D.x/ and E.x/ are real functions and fλng is a sequence of constants. If A.x/; C.x/; D.x/ and E.x/ are even functions, C.x/ is positive and B.x/ is an odd function then Z α Z α ∗ ∗ 0 if n 6D m; W x x x dx D W x 2 x dx with D (4) . /Φn. /Φm. / . /Φn . / δn;m δn;m D −α −α 1 if n m; where Z 0 Z ∗ B.x/ − A .x/ C.x/ B.x/ W .x/ D C.x/ exp dx D exp dx : (4:1) A.x/ A.x/ A.x/ Of course, the weight function W ∗.x/ must be positive and even on [−α; αU and x D α must be a root of the function Z B.x/ − A0.x/ Z B.x/ A.x/K.x/ D A.x/ exp dx D exp dx ; (4:2) A.x/ A.x/ i.e. A(α/K(α/ D 0. In this sense, since K.x/ D W ∗.x/=C.x/ is an even function then A.−α/K.−α/ D 0 automatically. By using this theorem fortunately many new symmetric orthogonal sequences can be obtained. For instance, a main class of symmetric orthogonal functions (MCSOF) has recently been introduced in [6]. Since we need the main properties of this class in the next sections let us restate them here. 2. Generation of MCSOF using Theorem 1.1 If the options A.x/ D x2.px2 C q/ evenI B.x/ D x.rx2 C s/ oddI C.x/ D x2 > 0 evenI (5) D.x/ D 0 evenI E.x/ D −θ.s C (θ − 1/q/ evenI for p; q; r; s 2 RI .−1/θ D −1; (θ/ n n and λn D − .n C (θ − 1/.1 − .−1/ /=2/.r C .n − 1 C (θ − 1/.1 − .−1/ /=2/p/ are substituted in the main equation (3) the second order differential equation 2 2 C 00 C 2 C 0 C (θ/ 2 − C − − − n D x .px q/Φn .x/ x.rx s/Φn.x/ λn x θ.s (θ 1/q/.1 . 1/ /=2 Φn.x/ 0; (6) will appear. According to [6], one of the basic solutions of this equation is a symmetric sequence of orthogonal functions in the form Tn=2]−1 n C C − − C 1−.−1/n (θ/ r s Y .2j 1 .1 . 1/ )θ/ q s . )(θ−1/ S x D x 2 n p q 2j − 1 C n C 1 − −1 n − 1 2 p C r jD0 . / )(θ = // Tn=2U Tn=2]−.kC1/ n ! X Tn=2U Y .2j − 1 C n C .1 − .−1/ )(θ − 1=2// p C r − × xn 2k; (7) k 2j C 1 C 1 − −1 n q C s kD0 jD0 . / )θ/ which satisfies the recurrence relation (θ/ r s 1C.−1/n(θ−1/ (θ/ r s (θ/ r s (θ/ r s S C x D x S x C C S − x ; (8) n 1 p q n p q n p q n 1 p q where r s C(θ/ n p q 1 C .−1/n(θ − 1/ pqn2 C (θ − 3 C 3.−1/n.1 − θ//pq C .1 C .−1/n(θ − 1//qr − .−1/nθps n C ..1 − θ/.p − r/q C ((θ − 3/p C r/s/ .1 − .−1/n/=2 D : (8:1) ...−1/n(θ − 1/ C 2n C θ − 2/p C r/...−1/n.1 − θ/ C 2n C θ − 4/p C r/ As the recurrence relation (8) shows, MCSOF is reduced to a main class of symmetric orthogonal polynomials (MCSOP) if and only if θ D 1. In other words, MCSOP satisfies the relation .1/ r s .1/ r s .1/ r s .1/ r s S C x D xS x C C S − x ; (9) n 1 p q n p q n p q n 1 p q in which r s pqn2 C ..r − 2p/q − .−1/nps/ n C .r − 2p/s.1 − .−1/n/=2 C.1/ D : (9:1) n p q .2pn C r − p/.2pn C r − 3p/ M. Masjed-Jamei, M. Dehghan / Computers and Mathematics with Applications 56 (2008) 2941–2947 2943 See [7] for more details. The functions (7) also satisfy a generic orthogonality relation as Z α r s (θ/ r s (θ/ r s D W x Sn x Sm x dx Nnδn;m; (10) −α p q p q p q where Z − 2 C − Z − 2 C r s 2 .r 4p/x .s 2q/ .r 2p/x s W x D x exp dx D exp dx ; (10:1) p q x.px2 C q/ x.px2 C q/ denotes the corresponding weight function and T U 2 n=2 n n Y .1/ r C .1 − .−1/ )θp; s C .1 − .−1/ )θq N D C n i p; q iD1 Z α n n r C .1 − .−1/ )θp; s C .1 − .−1/ )θq × W x dx; (10:2) −α p; q shows the generic value of the norm square and finally α takes the standard values 1; 1. (θ/ Although we introduced four main sub-classes of Sn .xI p; q; r; s/ in [6], further important sub-classes can still be found for the specific values of p; q; r; s and θ. In this paper, we introduce one of the mentioned samples, which firstly generalizes Fourier trigonometric sequences and is secondly orthogonal with respect to the same constant weight function on T0; πU. Here it may be interesting for the reader that there is also a special trigonometric sequence, which is orthogonal with respect to the normal distribution exp.−γ x2/ on .−∞; 1/ though it is not a special case of MCSOF. For more details, see [8]. To introduce a classical generalization of Fourier trigonometric sequences, we should first (for convenience) suppose that in equation (6) p D −1 and q D 1. In this case, by the change of variable x D cos t, the modified differential equation (6) is transformed to − C 3 − − − n 2 00 .r 1/ cos t s cos t 0 (θ/ 2 1 .