Mathieu-type Series for the aleph-function Occuring in Fokker-Planck Equation

Vol. 3, No. 6, 2010, 980-988

ISSN 1307-5543 – www.ejpam.com

S

I

C

A

: T

A

P

H

M. S

,

ON THE OCCASION OF HIS

70

Mathieu–type Series for the

ℵ

–function Occurring in

Fokker–Planck Equation

Ram K. Saxena

, Tibor K. Pogány

1_{Department of Mathematics and Statistics, Jain Narain Vyas University, Jodhpur–342004, India} 2_{Faculty of Maritime Studies, University of Rijeka, Studentska 2, 51000 Rijeka, Croatia}

Abstract. Closed form expressions are obtained for a family of convergent Mathieu typea–series and its alternating variant, whose terms contain an_ℵ–function, which naturally occurs in certain problems associated with driftless Fokker–Planck equation with power law diffusion[25]. The_ℵ–function is a generalization of the familiarH–function and theI–function. The results derived are of general char-acter and provide an elegant generalization for the closed form expressions the Mathieu–type series associated with theH–function by Pogány[8], for Fox–Wright functions by Pogány and Srivastava[13]

and for generalized hypergeometric _pF_q and Meijer’s G–function by Pogány and Tomovski[16], and others. For theH–function[7, p. 216] the results are obtained very recently by Pogány and Saxena

[11].

2000 Mathematics Subject Classifications: Primary 33C20, 33C60; Secondary 40G99, 44A20. Key Words and Phrases:I–function, Dirichlet series,H–function, Fox–Wright function, Laplace inte-gral representation for Dirichlet series, Mathieua–series, Mellin–Barnes type integrals, Mittag–Leffler function

1. Introduction and Preliminaries

In order to unify and extend the results for the convergent Mathieu–type a–series and its alternating variants whose terms contain the familiar transcendental functions, such as Gauss hypergeometirc function₂F₁, generalized hypergeometric function_pF_q, the Fox–Wright

∗_{Corresponding author.}

Email addresses:poganjpfri.hr(T. Pogány),ram.saxenayahoo.om(R. Saxena)

function, the Meijer’s G–function and Fox’s H–function published in a series of papers by Pogány[8, 9, 10], Pogányet al.[11, 12, 13, 14, 15, 16]and Srivastava and Tomovski[23].

Inequalities and integral representations for Mathieu–type series are discussed by Cerone and Lenard[1], Pogány and Tomovski[17], Srivastava and Tomovski[23]and others. The results obtained by the authors in this serve as the key formulæ for numerous potentially useful special functions of Science, Engineering and Technology scattered in the literature.

In the study of fractional driftless Fokker–Planck equations with power law diffusion co-efficients, there arises naturally a special function, which is a special case of the ℵ, that is Aleph–function. The idea to introduce Aleph–function belongs to Südlandet al. [24], how-ever the notation and complete definition is presented here in the following manner in terms of the Mellin–Barnes type integrals[also see 25]:

ℵ[z] =_ℵp m,n

i,qi,τi;r[z] =ℵ m,n pi,qi,τi;r

–

z

(a_j,A_j)_1,n, . . . ,[τ_j(a_j,A_j)]_n+1,pi (bj,Bj)1,m, . . . ,[τj(bj,Bj)]m+1,qi

™

:= 1 2πω

Z

L

Ω_p m,n i,qi,τi;r(s)z

−s_{ds (1)}

for allz6=0, whereω=p₋1 and

Ω_p m,n i,qi,τi;r(s) =

m

Q

j=1

Γ(b_j+B_js)· n

Q

j=1

Γ(1−a_j−A_js)

r

P

i=1

τ_i pi

Q

j=n+1

Γ(a_ji+A_jis)· qi

Q

j=m+1

Γ(1−b_ji−B_jis)

, (2)

The integration pathL =L_iγ∞,γ∈R _{extends fromγ}−_i∞_toγ+i∞_{, and is such that the} poles, assumed to be simple, of Γ(1−a_j −A_js),j = 1,n do not coincide with the poles of Γ(b_j+B_js),j=1,m. The parametersp_i,q_i are non–negative integers satisfying 0≤n≤p_i, 1≤m≤q_i,τ_i >0 fori=1,r. The parametersA_j,B_j,A_ji,B_ji>0 anda_j,b_j,A_ji,b_ji ∈C_{. The} empty product in (2) is interpreted as unity. The existence conditions for the defining integral (1) are given below:

ϕ_ℓ>0, |arg(z)|<π

2ϕℓ ℓ=1,r; (3)

ϕℓ≥0, |arg(z)|<

π

2ϕℓ and ℜ{ζℓ}+1<0 , (4) where

ϕℓ=

n

X

j=1

Aj+ m

X

j=1

Bj−τℓ Xpℓ

j=n+1

Ajℓ+

qℓ X

j=m+1

Bjℓ

(5)

ζ_ℓ= m

X

j=1

b_j− n

X

j=1

a_j+τ_ℓ qℓ X

j=m+1

b_jℓ− pℓ X

j=n+1

a_jℓ+1

2 pℓ−qℓ

Remark 1. If the sum in the denominator of(2)can be simplified in terms of a polynomial in s, the factors of this polynomial can be expressed by a fraction of Euler’s Gamma function leading to an H–function instead, see[25, p. 325].

Remark 2. It is observed that there is no historical name given to (1), compared to [24]. The Mellin transform of this function is the coefficient of z−s in the integrand of (1). There are no references containing tables ofℵ–functions in the literature.

Forτ₁=τ₂=. . .=τ_r=1, in (1) the definition of followingI–function[21]is recovered:

I[z] =_ℵp m,n

i,qi,1;r[z] =ℵ m,n pi,qi,1;r

–

z

(aj,Aj)1,n, . . . ,(aj,Aj)n+1,pi (b_j,B_j)_1,m, . . . ,(b_j,B_j)_m+1,q i

™

:= 1 2πω

Z

L

Ω_p m,n i,qi,1;r(s)z

−s_{ds, (7)}

where Ω_pi,qim_,1;,n_r(s) is defined in (2). The existence conditions for the integral in (7) are the same as given in (3)–(6) withτ_i=1,i=1,r.

If we further set r = 1, then (7) reduces to the familiar H–function given e.g. in the monograph[7]:

Hm_p,q,n[z] =_ℵp m,n

i,qi,1;1[z] =ℵ m,n pi,qi,τi;1

–

z

(a_p,A_p) (b_q,B_q)

™

:= 1 2πω

Z

L

Ω_p m,n i,qi,1;1(s)z

−s_{ds, (8)}

where the kernel Ω_p m,n

i,qi,1;1(s) is given in (2), which itself is a generalization of Meijer’s G– function[2, p. 207]to which it reduces for A₁ = . . .=A_p =1= B₁ =. . . = B_q. A detailed and comprehensive account of the H–function is available from the monographs written by Mathai and Saxena[6], Srivastavaet al. [22], Kilbas and Saigo[4]and Mathaiet al. [7].

In what follows, the Aleph function will be represented by the contracted notations ℵ m,n

pi,qi,τi;r[z]orℵ[z].

Now, consider the Mathieu–typea–seriesΘ_λ,µ and its alternating variantΘe_λ,µ, defined by

Θ_λ,µnℵ;c,xo:= ∞

X

j=1

ℵ m,n+1

pi+1,qi,τi;r

h_x

c_j

(α,₍β_b), (aj,Aj)1,n,(aji,Aji)n+1,pi

j,Bj)1,m, (bji,Bji)m+1,qi

i

cλ_j(c_j+x)µ , (9)

e

Θ_λ,µnℵ;c,xo:= ∞

X

j=1

(₋1)j−1_ℵ m,n+1

pi+1,qi,τi;r

h_x

c_j

(α,₍β_b), (aj,Aj)1,n,(aji,Aji)n+1,pi

j,Bj)1,m, (bji,Bji)m+1,qi

i

cλ_j(c_j+x)µ (10)

where the convention is followed that the positive sequence c = cn

n∈N monotonously

in-creases and tends to infinity; equivalently

2. Integral Representations of

Θ

n

ℵ

;

c

,

x

o

and

Θ

e

n

ℵ

;

c

,

x

o

The Laplace transform of theℵ–function can be established in the following form

Z ∞

0

xλ−1e−s x_ℵp m,n i,qi,τi;r

ηxρdx

=s−λℵ_pi+m_1,,n_qi+_,τ1 i;r

h_η

sρ

(1−₍λ_b,ρ), (aj,Aj)1,n, [τi(aji,Aji)]n+1,pi

j,Bj)1,m, [τi(bji,Bji)]m+1,qi

i

, (12)

whereλ,s,η∈C_;ℜ{_s}_{>0,ρ >0,τi>0,i=1,r, and}

ℜ{λ_}+ρ min

1≤j≤m ℜ{bj}

B_j >0, |arg(η)|< π

2 1min≤ℓ≤r ζℓ

; (13)

the parameterζℓ is defined in (6).

The formula (12) can be easily established with the help of the definition (2) ofℵ–function and using gamma function formula

Γ(µ)ζ−µ=

Z _∞

0

xµ−1e−ζxdx min

ℜ{µ},ℜ{ζ}

>0 .

Theorem 1. Letλ >0,µ >0,x >0,α= 1−λ,β = ρ and let the sequencec satisfies (11). Then there hold the following results:

Θ_λ,µn_ℵ;c,x

o

=Iℵ

c(λ+1,µ) +µIℵc(λ,µ+1) (14)

e

Θ_λ,µn_ℵ;c,x

o

=eIℵ

c(λ+1,µ) +µeIℵc(λ,µ+1), (15)

where

Iℵ

c(u,v):=

Z _∞

c1

c−1(t) tu_(t+x)v ℵ

m,n+1

pi+1,qi,τi;r

h_x

t;u

i

dt, (16)

e

Iℵ

c(u,v):=

Z ∞

c1

sin2 π₂[c−1(t)] tu(t+x)v ℵ

m,n+1

pi+1,qi,τi;r

h_x

t;u

i

dt, (17)

and

ℵ m,n+1

pi+1,qi,τi;r

h_x

t;u

i

:=ℵ_pi+m_1,,n_qi+_,τ1 i;r

h_x

t

(1−u, 1), (aj,Aj)1,n,[τi(aji,Aji)]n+1,pi (b_j,B_j)_1,m, [τ_i(b_ji,B_ji)]_m+1,qi

i

.

Proof. Takingζ=c_n+x in (14), settings=c_j;ρ=1,η=x and insertingα=1−λ,β=1 in (12), we find that

Θ_λ,µn_ℵ;c,x

o

= ∞

X

j=1

ℵ m,n+1

pi+1,qi,τi;r

h_x

cj

(1−λ, 1),(aj,Aj)1,n,[τi(aji,Aji)]n+1,pi (b_j,B_j)_1,m, [τ_i(b_ji,B_ji)]_m+1,qi

i

cλ_j(cj+x)µ

= 1

Γ(µ) ∞

X

j=1 Z _∞

0

sλ−1e−cjsℵ m,n

pi,qi,τi;r[x s]ds

Z _∞

0

tµ−1e−(cj+x)t_dt

= 1

Γ(µ)

Z ∞

0 Z ∞

0

‚_X∞

j=1

e−cj(s+t)

Œ

sλ−1tµ−1e−x tℵ_p m,n

i,qi,τi;r[x s]dsdt, (18) where, by convergence reasonsµ >0 is already assumed. Following the lines of the use of Dirichlet series technique used in earlier papers by Pogány and coworkers[10, 11, 12, 13, 14, 15, 16], by means of (12) we conclude

Θ_λ,µnℵ;c,xo= 1 Γ(µ)

Z ∞

0 Z ∞

0 Z ∞

c1

sλtµ−1e−(y+x)t−ysℵ_pi,qim_,,_τn i;r[x s]

c−1(y)dsdtdy

+ 1

Γ(µ)

Z ∞

0 Z ∞

0 Z ∞

c1

sλ−1tµe−(y+x)t−ysℵ_pi,qim_,,_τn i;r[x s]

c−1(y)dsdtdy :=J_s+J_t.

Introducing the auxiliary integral

Iℵ

c(u,v):=

Z _∞

c1

c−1(t) tu_(t+x)vℵ

m,n+1

pi+1,qi,τi;r

hx t;u

i

dt,

it readily follows that

Js=Iℵc(λ+1,µ) and Jt=µ·Iℵc(λ,µ+1).

This finishes the proof of (16).

The proof of (17) is similar to that of (16), if we employ the definition of the new alter-nating inner Dirichlet series_De_c(_·)[14, p. 77, Section 4]given below:

e

Dc(s+t) =

∞

X

j=1

(−1)j−1e−cj(s+t)_{= (s+t)}

Z _∞

c1

e−(s+t)xsin2π 2

c−1(x)dx. (19)

The application of (19) completes the proof of (17).

3. Special Cases

Whittaker functions, hypergeometric functions, generalized hypergeoemtric_pF_qfunction, Mei-jer’s G–function, Fox–WrightΨ function and Fox H–function and their special cases can be deduced by making suitable changes in the parameters. But, for the sake of brevity, some interesting special cases of Theorem 1 are given below.

Corollary 1. [12, Theorem]Letλ,µ,x > 0,α=1−λ,β =ρ,τ₁ =. . .=τ_r =1, and let the sequence csatisfies(11). Then the Aleph function reduces to an I–function and there holds the following result

Θ_λ,µnI_pm,n+1 i+1,qi,r;c,x

o

=II

c(λ+1,µ) +µI

I

c(λ,µ+1) (20)

e

Θ_λ,µnI_pm,n+1 i+1,qi,r;c,x

o

=eII

c(λ+1,µ) +µeI

I

c(λ,µ+1), (21)

where

II

c(u,v):=

Z ∞

c1

c−1(t) tu(t+x)v I

m,n+1

pi+1,qi,r

h_x

t;u

i

dt, (22)

eII

c(u,v):=

Z ∞

c1

sin2 π

2[c

−1_(t)]

tu_(t+x)v I m,n+1

pi+1,qi,r

h_x

t;u

i

dt, (23)

with

I_pim₊,n_1,+_qi1_,rhx t;u

i

:=ℵ_pi+m_1,,n_qi+_,1;1_rhx t

(1−u, 1), (aj,Aj)1,n,(aji,Aji)n+1,pi (bj,Bj)1,m, (bji,Bji)m+1,qi

i

. Here c remains the same as above in Theorem 1.

When r=1,τ1=1, the Aleph function reduces to Fox’sH–function and Theorem 1 gives

rise to the following result given by Pogány[10, Theorem].

Corollary 2. Letλ,µ,r>0,α=1−λ,β=ρ=1and let the sequencecsatisfies the condition given in(11). Then we have

Θ_λ,µnH_pm₊,n_1,+_q1;c,x

o

=IH

c(λ+1,µ) +µI

H

c(λ,µ+1) (24)

e

Θ_λ,µnH_pm₊,n_1,+_q1;c,xo=eIH

c(λ+1,µ) +µeI

H

c(λ,µ+1), (25)

where

IH

c(u,v):=

Z ∞

c1

c−1(t) tu(t+x)v H

m,n+1

p+1,q

h_x

t

(1−u_(b, 1),(aj,Aj)1,n,(aj,Aj)n+1,p

j,Bj)1,m,(bj,Bj)m+1,q

i

dt (26)

e

IH

c(u,v):=

Z ∞

c1

sin2 π

2[c

−1_{(t) ]}

tu(t+x)v H m,n+1

p+1,q

h_x

t

(1−u₍, 1),_b (aj,Aj)1,n,(aj,Aj)n+1,p

j,Bj)1,m,(bj,Bj)m+1,q

i

dt. (27)

The Fox–Wright functionpΨqis defined[7, p. 23]by the power series in the form

pΨq

h _(a

p,αp) (bq,βq)

z i := ∞ X

n=0 Qp

j=1Γ aj+Ajn

Qq

j=1Γ bj+Bjn

z

n

with a_j,b_j ∈C_,Aj,Bj ∈R_,Ai·_Bj 6_{=0(i=1,p,j=1,q)and}Pq j=1Bj−

Pp

j=1Aj >−1. Now, employing the identity[7, p. 25]

pΨq

h _(a

p,Ap) (b_q,B_q)

−z

i

=H_p1,_,qp₊₁

h

z

_{(0, 1),}(1−₍₁a₋p,A_bp)

q,Bq)

i

, (29)

pointing out that we can express itviathe Aleph function as

pΨq

h _(a

p,Ap) (bq,Bq)

−z

i

=_ℵp,q1,₊p_1,1;1hz

_{(0, 1),}(1−₍₁a₋p,A_bp)

q,Bq)

i

, it is not difficult to deduce the corresponding results for Fox–Wright function.

Let us define

Θ_λ,µp+1Ψ_q;c,x := ∞

X

j=1

p+1Ψq

h _(α,β),(a

p,Ap) (bq,Bq)

−_cx

j

i

cλ_j(c_j+x)µ , (30)

and

e

Θ_λ,µp+1Ψ_q;c,x := ∞

X

j=1

(₋1)j−1_p+1Ψ_qh (α,β),(ap,Ap) (b_q,B_q)

− _cx

j

i

cλ_j(c_j+x)µ . (31)

We then obtain the following

Corollary 3. Letλ_6∈N_{,µ >0,r>0,(α,β) = (1}−_{λ, 1),(bq,βq) = (1, 1)and let the sequencec}

satisfies(11). Then we have

Θ_λ,µp+1Ψ_q;c,r =IΨ

c(λ+1,µ) +µI

Ψ

c(λ,µ+1) (32)

e

Θ_λ,µp+1Ψ_q;c,r =eIΨ

c(λ+1,µ) +µeI

Ψ

c(λ,µ+1), (33)

where

IΨ

c(u,v):=

Z ∞

c1

c−1(t)

tu_(t+x)v ·p+1Ψq

h _{(1−u, 1),(a}

p,Ap) (1, 1),(b_q−1,B_q−1)

− x_t

i

dt (34)

and

e

IΨ

c(u,v):=

Z _∞

c1

sin2 π₂[c−1(t) ]

tu_(t+x)v · p+1Ψq

h _{(1−u, 1),(a}

p,Ap) (1, 1),(b_q−1,B_q−1)

− x_t

i

dt. (35) Remark 3. Finally, it is interesting to observe that by virtue of the relation

Eα,β(z) =H1,21,1 h

−z

_{(0, 1),}(0, 1)_(1−β,α) i

=_ℵ1,2,1;11,1 h₋z

_{(0, 1),}(0, 1)_(1−β,α) i

, where E_α,β(z)is the Mittag–Leffler function[Chapter 18 3]and[p. 80 5], defined by

Eα,β(z) =

∞

X

n=1

zn Γ(αn+β),

whereα,β ∈C_;ℜ{_α}_,ℜ{_β}_{>0, the similar type of results for the Mittag–Leffler function can}

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