A MATLAB program for the computation of the confluent hypergeometric function Φ2

1 A MATLABTM program for the computation of the confluent hypergeometric function Φ2 E. Martos-Naya, J. M. Romero-Jerez, F. J. Lopez-Martinez and J. F. Paris Abstract We here present a sample MATLABTM program for the numerical evaluation of the confluent hypergeometric function Φ2. This program is based on the calculation of the inverse Laplace transform using the algorithm suggested by Simon and Alouini in their reference textbook [1]. I. THE MULTIVARIATE Φ2FUNCTION The following confluent form of the generalized Lauricella series is defined in [2, eq. 7.2, pp. 446] Φ(n) 2(b1, . . . , bn;c;x1, . . . , xn), ∞ X m1...mn (b1)m1. . . (bn)mn (c)m1+...+mn xm1 1 m1!. . . xmn n mn!(1) where (·)mdenotes the Pochhammer symbol. This function, also regarded as confluent hypergeometric function of nvariables [3], makes appearances in numerous problems in communication theory [4–8], either in a bivariate form (n= 2) or in a multivariate fashion. Because it is defined as an n-fold infinite summation, its numerical evaluation poses some challenges from a computational point of view. However, the Laplace transform of the Φ2function has a comparatively simpler form, in terms of a finite productory of elementary functions. Specifically, in [9, 4.24.5], the following Laplace transform pair is listed: Lntc−1Φ(n) 2(b1, . . . , bn;c;x1t, . . . , xnt) ; t, so=Γ(c) sc1−x1 s−b1 . . . 1−xn s−bn ,(2) which is valid for <(c)>0,<(s)>0,bi∈R,i= 1 . . . n. Thus, we have that Φ(n) 2(b1, . . . , bn;c;x1, . . . , xn) = tc−1Φ(n) 2(b1, . . . , bn;c;x1t, . . . , xnt)|t=1 =L−1Γ(c) sc1−x1 s−b1 . . . 1−xn s−bn ;s;t(3) 2 Therefore, the Φ2function can be evaluated by means of an inverse Laplace transform. Due to numerous requests, we here provide a MATLABTM sample code for the evaluation of the Φ2 function. This code implements an Euler summation-based technique described in detail in the appendix 9B of the reference textbook by Simon and Alouini [1], inspired in [10]. II. MATLABTM CODE 1%% y=Phi2 ( b vec to r , c , xvect or ,N) 2% 3% b v e c t o r : [ b1 . . . bn ] 4% c : s c a l a r p a r a m e t e r Re ( c )>0 5% x v e c t o r : [ x1 . . . xn ] 6% N : T r u n c a t i o n of i n f i n i t e summation 7% 8function y=Phi2 ( b vec to r , c , xve ct or ,N) 9% H e u r i s t i c a l l y a d j u s t e d so t h a t t h e d i s c r e t i z a t i o n e r r o r term 10 % abs {E(A)}<exp(−A) 11 A=15; 12 % I n v e r s e La place Transform 13 K=exp (A/ 2 ) ; 14 a l p h a i n v = [ 0 . 5 , ones ( 1 ,N−1) ] ; 15 n =0:N−1; 16 y1 =( −1) . ˆ n . ∗alphainv .∗r e a l ( L aplac eP hi 2 ( (A+2∗p i ∗1 i ∗n ) / 2 , c , x ve ct or , b v e c t o r ) ) ; 17 y=K. ∗sum ( y1 ) ; 18 19 function y= L apla cePhi2 ( s , c , x , b ) 20 P=ones ( 1 , length( s ) ) ; 21 f o r k =1: length ( b ) 22 P=P .∗((1 −x ( k ) ∗( s . ˆ ( −1 ) ) ) .ˆ( −b ( k ) ) ) ; 23 end ; 24 y=gamma ( c ) ∗( s .ˆ( −c ) ) . ∗P ; 3 REFERENCES [1] M. K. Simon and M.-S. Alouini, Digital communication over fading channels. Wiley-IEEE Press, 2005. [2] A. Erd´ elyi, Beitrag zur theorie der konfluenten hypergeometrischen funktionen von mehreren ver¨ anderlichen. H¨ olderPichler-Tempsky in Komm., 1937. [3] P. W. K. H. M. Srivastava, Multiple Gaussian Hypergeometric Series. John Wiley & Sons, 1985. [4] D. Morales-Jimenez and J. F. Paris, “Outage probability analysis for η-µfading channels,” IEEE Communications Letters, vol. 14, no. 6, pp. 521–523, June 2010. [5] N. Zlatanov, Z. Hadzi-Velkov, and G. K. Karagiannidis, “An efficient approximation to the correlated Nakagami-m sums and its application in equal gain diversity receivers,” IEEE Transactions on Wireless Communications, vol. 9, no. 1, pp. 302–310, January 2010. [6] J. F. Paris, “Closed-form expressions for Rician shadowed cumulative distribution function,” Electronics Letters, vol. 46, no. 13, pp. 952–953, June 2010. [7] S. Kalyani and R. M. Karthik, “The Asymptotic Distribution of Maxima of Independent and Identically Distributed Sums of Correlated or Non-Identical Gamma Random Variables and its Applications,” IEEE Transactions on Communications, vol. 60, no. 9, pp. 2747–2758, September 2012. [8] J. F. Paris, “Statistical Characterization of κ-µShadowed Fading,” IEEE Trans. Veh. Technol., vol. 63, no. 2, pp. 518–526, Feb 2014. [9] A. Erd´ elyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. [10] J. Abate and W. Whitt, “Numerical inversion of Laplace transforms of probability distributions,” ORSA Journal on computing, vol. 7, no. 1, pp. 36–43, 1995.