Some representations on multiindices unified Voigt functions.

1. Introduction

To consider the importance of the Voigt functions occuring rather frequently in the wide variety of problems in spectroscopy, neutron physics and several other devices in the field of physics. A number of authors namely Srivastava et al. [12], Srivastava and Miller [11], Klusch [4], Aromstrong and Nicholls [1], Reiche [7] and others have studied various mathematical properties for the Voigt functions and their generalizations.

Recently M.Kamarujjama et al [3] have studied on multiindices representations of unified Voigt functions in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)

(x, z, y [member of] [R.sup.+], and Re([mu] + [n.summation over (i=1)] [v.sub.i]) > -1, (v) = [v.sub.1], ***, [v.sub.n] [member of] [R.sup.n]w) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the Hyper-Bessel function of order n, defined by (see (Delerue [2]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Evidently,

[[ohm].sub.[mu],(-1/2)] [x, y, z] = [[K.sub.[mu]+n/2] [x,y,z], [[ohm].sub.[mu],(1/2)] [x,y,z] = [L.sub.[mu]+n/2] [x,y,z]} (1.3)

where v =([??] 1/2) = ([??] 1/2, [?? ]1/2, ... ,[??] 1/2) [member of] [R.sup.n].

For z = 1/4, integral (1.1) is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

It is not difficult to observe that when n = 1, equation (1.3) and (1.5) reduced to results of Klusch [4] and Srivastava and Miller [11] respecttively.

2. New Representations of [[ohm].sub.[mu],(v)][x, y, z]

For the purpose of the present study, we recall the definition of generalized Lauricella function (see Srivastava and Daoust [8]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

A detailed discussion of the conditions of convergence of the multiple series (2.1) is given in the paper of Srivastava and Daoust [9].

We shall obtain the following new explicit multiindices representation for the unified Voigt function defined by (1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

where [F.sup.(2)] is a special case of Srivastava-Daoust function [8;p.454] for r=2.

The above results can be easily proved by a mathod similar to applied by Srivastava et al.[12] for obtaining explicit expression of the generalized Voigt functions.

For (v) = (-1/2) and (v) = (1/2), equation (2.2) reduces to multiindices representations of the generalized Voigt functions [K.sup.[mu]][x, y, z] and [L.sub.[mu]][x, y, z] of Klusch [4] in the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

where [mu], x, z, y [member of] [R.sup.+].

For n = 1, equation (2.3) and (2.4) reduce to known results of Srivastava et al [12].

3. Partly Bilateral and Partly Unilateral Representations

We start with a known result, due to Pathan and Yasmeen [5] in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

(where [m.sup.*] = max{0, -m}), replacing s [right arrow] [su.sup.2], t [right arrow] [tu.sup.2] and p [right arrow] [pu.sup.2] and then multiplying both sides of (3.1) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and integrating with respect to u from zero to infinity and using the integral representation (1.1), we thus obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

where w = (z - s - t + pt/s) and [alpha] = 1/2([mu] + [summation] [v.sub.i] + 1).

Now separeting the q-series into even and odd terms, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

If we take (v) = ([??] 1/2), in equation (3.3), we get (cf. equation (1.3))

which is the valid under the same conditions as mentioned by (3.2) and [F.sup.(3)] is a special case of the Srivastava Daoust functions [8,p.454] for r = 3.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

where c = 1/2 ([mu] + 1), respectively.

For n = 1, equations (3.3), (3.4) and (3.5) reduce to known results of Srivastava et al. [12] and s = t = p/2 equation (3.3) gives the partly bilateral and partly unilateral representation of the multiindices unified Voigt functions (2.2).

For n = 1 and [mu] = 1/2 along with s = t = p/2 and z = 4, equations (3.4) and (3.5) reduce to the representation of K(x, y) and L(x, y), respectively.

4. A set of Expansions

On expanding the left member of (3.3) and using the representation (2.2), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

where w = (z - s - t + tp/s) and [alpha] = 1/2([mu] + [summation] [v.sub.i], + 1).

On setting y = 0, the relation (4.1) reduce to the following generating function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

where [sub.p][psi][sub.q] is the generalized hypergeometric function defined by [10;p.50(21)]

On setting [v.sub.1]0 = [v.sub.2] = *** = [v.sub.n] = 0 and x = 0 in (4.1), we obtain the following interesting relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

where c = 1/2 ([mu] + 1] and [sub.p][[psi].sub.q] is the generalized hypergeometric function defined by [10,p.50(21)].

For n = 1 [member of] [R.sup.1], equation (4.1) to (4.3) reduce to known results of Srivastava et al. [12].

References

[1] B.H. Armstrong and R.W. Nicholls, Emission, Absorption and Transfer of Radiation in Heated Atmospheres, Pergamon Press, New York, 1972.

[2] P. Delerue, Ser la calcul symboliqueanvariables et Sur les functions hyper bess eliennes I and II, Ann. Soc. Sci. Bruxeles, 67(1):83-104 and 229-274, 1953.

[3] M. Kamaruujama, M. Khursheed Alam and D. Singh, New Results concerning the Analysis of Voigt Functions, South East Asian J. Math and Math. Sci., 3(1):93- 102, 2004.

[4] D. Klusch, Astrophysical Spectroscopy and neutron reactions, Integral transforms and Voigt functions, Astrophys. Space Sci., 175:229-240, 1991.

[5] M.A. Pathan and Yasmeen, On partly bilateral and partly unilateral generating functions, J. Austral. Math. Soc. Ser. B, 28:240-245, 1986.

[6] E.D. Rainville, Special functions, The Macmillan Company, New York, 1960.

[7] F. Reiche, Uber die Emission, Absorption and Intensitatsvertcitung Von spektrallicien, Ber. Deutsch. Phys. Ges., 15:3-21, 1913.

[8] H.M. Srivastava and M.C. Daoust, Certain generalized neumann expansions associated with the Kampe de Feriet function, Nederl. Akad. Wetench. Proc. Ser. A72=Indag. Math., 31:449-457, 1969.

[9] H.M. Srivastava and M.C. Daoust, A note on the convergence of Kampe de Feriet doubles hypergeometric series, Math. Nachr., 53:152-159, 1972.

[10] H.M. Srivastava and H.L. Manocha, A tretise on generating functions, Halsted Press (Ellis Horwood Ltd., Chichester) 1984.

[11] H.M. Srivastava and E.A. Miller, A unified presentation of the Voigt functions, Astrophys. Space Sci., 135:111-118, 1987.

[12] H.M. Srivastava, M.A. Pathan and M. Kamarujjama, Some unified presentations of the generalized Voigt functions, Comm. Appl. Anal., 2:49-64, 1998.

M. Kamarujjama

Department of Applied Mathematics, Z.H. College of Engineering and Technology Aligarh Muslim University, Aligarh-202002, India.

E-mail: mdkamarujjama@rediffmail.com

Dinesh Singh

M. A. Inter College, Badhon, Aligarh-202001 (U.P), India.

E-mail: dineshamu@yahoo.com

M. Ghayasuddin

Department of Applied Mathematics, Z.H. College of Engineering and Technology Aligarh Muslim University, Aligarh-202002, India.

E-mail: ghayas.maths@gmail.com
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