Some results on integrals involving generalized Jacobi and related functions

Computers Math. Applic. Vol. 30, No. 1, pp. 47-52, 1995

Pergamon

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S o m e R e s u l t s on Integrals I n v o l v i n g G e n e r a l i z e d Jacobi and R e l a t e d F u n c t i o n s R . SRIVASTAVA Department of Mathematics and Statistics, University of Victoria Victoria, British Columbia V8W 3P4, Canada and Centre for Mathematical Sciences Trivandrum 695014, Kerala, India REKHAS©UVVM. UVIC. CA

(Received November 1994; accepted January 1995) A b s t r a c t - - M o t i v a t e d by several recent works on integrals involving various orthogonal polynomials and the natural logarithmic function, we consider a general integral Ia,v,m b (Z; A, #), defined by equation (1.1) below, and its partial derivatives with respect to the parameters a and b. The kernel S~(z) of these integral formulas is a general class of functions which stem essentially from the polynomials considered, over two decades ago, by Srivastava [1]. We discuss numerous applications of our main results involving familiar special functions and polynomials. We also give a simple proof of an identity involving the Psi (or Digamma) function. K e y w o r d s - - O r t h o g o n a l polynomials, Psi (or digamma) function, Gauss quadrature formulas, Special functions, Pfaff-Saalsch/itz theorem, Euler transformation.

1. I N T R O D U C T I O N I n our present investigation, we consider the following integral: ,b ~ ; A , # ) - - fl ( 1 - - X ) a ( l + x ) b S , m ( z ( 1 - - X ) ~ ( l + x ) ' ) , Ia~'mrz 1

(1.1)

where

Sin(z) = ~

k(-V)mk ~ A,,k z k,

(1.2)

k=0 m C N = { 1 , 2 , 3 , . . . }, v is a complex n u m b e r , the coefficients

Av,k (k E No = N U {0}) are

b o u n d e d , a n d (A)n denotes the well-known P o c h h a m m e r symbol defined by

(~)~ _

C(A+n)

F(A)

_ f 1,

n----0,

~. A(A+I)-..(A+n-1),

VneN.

(1.3)

Obviously, for v = n (n e N0), the series in (1.2) will t e r m i n a t e at k = [n/m]. T h e p o l y n o m i als Sin(z) (n E No) were introduced, over two decades ago, by Srivastava [1]. The present investigation was carried out at the Centre for Mathematical Sciences in qYivandrum where the author was a Visiting Professor and Director-in-Charge during the first quarter of 1994. The author wishes to thank A. M. Mathai of McGill University at Montr@al for the facilities provided at the Centre for Mathematical Sciences in Trivandrum during the preparation of this paper. Typeset by AA4~-TEX 47

48

R. SRIVASTAVA

V~77]* . In this paper, we evaluate the partial derivatives of Ia, b (z, A, #) with respect to the parameters a and b and connect our results to many known results scattered throughout the literature [2-6] involving integrals of orthogonal polynomials and the natural logarithmic function. Such integrals are useful in many scientific fields; for example, they are useful in Gauss quadrature formulas with algebraic and logarithmic singularities. The coefficients AL,,k in (1.2) are very general in nature; therefore, by suitably specializing these coefficients, Sin(z) can be reduced to various familiar functions and polynomials including, for example, those associated with Hermite, Jacobi, Laguerre, Gegenbaner, Legendre, and Chebyshev (see, for details, [7]). Other interesting special cases of the function Sin(z) include such generalized hypergeometric polynomials as the Bessel polynomials [8], the Gould-Hopper polynomials [9], and also the (substantially more general) Brafman polynomials [10]. Furthermore, our integrals involving S~ (z) will apply also to such other hypergeometric polynomials as the extended Jacobi polynomials and their generalizations studied in the literature (cf., e.g., [11,12]). In view of the generality of the function defined by (1.2), and of the frequent requirement of various results involving many of their aforementioned special cases, we find it worthwhile to evaluate the integral (1.1) and consider its partial derivatives as mentioned above.

2. M A I N

INTEGRAL

FORMULAS

By appealing to the definition (1.2) and the familiar (Eulerian) Beta integral [13, p. 9, equation 1.5(3)]

/1

(1 - x) ~ (1

+

x) b dx = 2 ~+b+l

1

/0

P(1

-

t) b dt (2.1)

= 2a+b+l r(a + 1)r(b + 1) r ( a + b + 2) '

~(a) > -1;

~(b) > -1,

it is not difficult to evaluate the integral on the right-hand side of (1.1). We thus find that

Ia,V~m b (z;A,/z)= ~

( 1 - x)a (l + x) b Sm ( z ( 1 - x))~(l % x) ~') dx 1

oo (-v)mk r ( a + Ak + 1)r(b + ~k + 1) , (2~+ , z)k, = 2~+5+1 E k! r (a + b + (A +/~)k + 2) A~ k k=0 N(a+Ak) > - 1 ;

N(b -{-#k) > - l ,

(2.2)

kEN0,

provided that the series on the right-hand side of (2.2) converges absolutely. Now, starting from the integral formula (2.2) and using logarithmic differentiation, we obtain our first main result given below. Integral Formula I D

Ja,b (z, A, ~) = N I°'b (z, A, .) _' {ln(1

x)}(1 - x)a(1 + X) b Svm(z(1 - z)A(1 + x)") dx

1

= Ia, b (z; A, #) In 2

(2.3)

oo + 2~+b+~ ~ (-u)~k r(a + ,xk + 1) r(b + ~k + 1) A,,,k (2~+"z)k k! r(a + b + (,~ + , ) ~ + 2) k=O

× {~(~ + xk + 1) - ~o(~ + b + (a + , ) k + 2) }, ~R(a+Ak) > - 1 ;

~R(b+#k)>-l,

kENo,

Generalized Jacobi and Related Functions

49

where ¢(z) denotes the Psi (or Digamma) function defined by

¢(z) =

d

r'(z) { l o g r ( z ) } - r(z)'

(2.4)

which does indeed possess the following well-known (rather classical) property: oo

(~)n

- ¢(~) - ¢ ( e - ~),

~(~ - ~) > 0;

e # o,-:,-2,.

n=1 n03)n

(2.~) "'"

In precisely the same manner as indicated above, we find from (2.2) our second main result given below. Integral Formula II

Ka, b (z; A, #) = ~-~ Ia, b (z; A, #) =

{ln(l+x)}(1-x)~(l+x)bS~(z(1-x)~(l+x)

") dx

1 ~m

(2.6)

= I~, b (z; ,~, #) In 2 +

2a+b+l

~-, (-u)~k r(a + ),k + 1) r(b + .k + 1) k! F(a + b + (A +/~)k + 2) A,,k (2A+"z) k k=O

x {~p(b+ #k +

I)

- ¢(a + b + (A + #)k + 2)},

~(a+Ak) >-1;

~(b+#k) >-1,

kGN0.

The following results can easily be deduced from our integral formulas (2.3) and (2.6). Integral Formula III

La,V ibm (z,• A, #) = J::'~(z; A, #) + ga,V,~T~ b (z; A, #) =

f_

{ln(1-x2)}(1-x)~(l+x)bS~(z(l-z)~(l+z)")dx

= 2I:'r2(z;a,.)

÷

ln2

(2.7)

oo (-v)mk r(a + ),k + 1) r(b + ~k + :) k! F(a + b + ()~ + #)k + 2) Av,k (2~+~z) k

2a+b+l ~

k=0

x {~b(a+Ak+l)+@(b+#k+l)-2@(a+b+(A+#)k+2)}, ~(a+)~k)>-l;

~(b+#k)>-l,

kEN0.

Integral Formula IV

Ma, b (z; ~, #) = J,~,b (Z, ~, #)-- K,, b (z; ,~,#) =

/_.((,.)} In ~

(1-x)a(l+x)bSm(z(1-x)~(l+x)~')dx

1

2a+b+l ~ , (-v)mk r(a + ),k + 1) r(b + #k + 1) k! r ( a + b + (~ + #)k + 2) A~,,k (2~+~z) k k=0

× {~;(a + Ak + 1) - ~b(b+ #k + 1)}, ~(a+~k) >-1; ]O-I-E

~(b+#k) >-1,

kEN0.

(2.s)

50

R. SRIVASTAVA

A unification (and generalization) of the integral formulas (2.3), (2.6), (2.7), and (2.8) is provided by the following result: Dv,m'P(

~,. X v,m u,m "°o,b,~ ,-,-., ~) = P Ja,b (z; A, ~) + a Ka, b (z; A, #)

=

{ln [(1

-z)°(l+x)~']}(l-x)~(l+z)bS~(z(l-x)~(l+z)")dx

1

= (p + a) I~, b (z, a, #) In 2

(2.9)

(-u).~k r(a + Ak + 1) r(b + ~k + 1) k! r ( a + b + ()~ + #)k + 2) Av,k (2"X+Pz) k

+ 2a+b+l k=0

x {pg/(a+Ak+l)+a¢(b+#k+l)-(p+a)¢(a+b+(A+#)k+2)}, ~ ( a + Xk) > - 1 ;

~ ( b + # k ) > -1,

k E No;

- c o < p < co;

- o o < a < co.

Clearly, we have ~,~'r~,l /

tta,b, o LZ, ~,#) = Ja,b (z,)~, #);

.

(2.10)

Ka, b (z; ~,/./,);

(2.11)

l"ta,b, 1

.

V,~

(Z;)~,~)=

RU,m,

a,b,11 /t z;.~,p.) = La,v , mb (z;A,#),

(2.12)

and RV,m,1 [ a,b,_l(Z;~,l£)

3. A P P L I C A T I O N S

:

Ma,y , mb (z,)t, #).

(2.13)

.

TO SPECIAL

FUNCTIONS

By appropriately specializing the coefficients A~,k, and by suitably choosing the parameters ~, #, and v, each of our integral formulas of the preceding section can be applied to yield the corresponding result for various families of special functions and polynomials. For example, if we set P r(a+~+v+~k+l) 1-I r ( ~ j + ~ k ) A~'k=( a+v)v q j=l , (3.1) r(a+vk+l) 11 r @ + n j k ) j=l

where ~>0,

~>0,

~>0,

and

q

j=l,...,p,

~>0,

j=l

.... ,q,

P

(r/- ~) + E

'r/j - E

j=l

~j >_ m - 1,

mEN,

j=l

each of our integral formulas (2.2), (2.3), (2.6), (2.7), and (2.8) will yield the corresponding result involving Wright's generalized hypergeometric function (see, e.g., [13, p. 183] and [14, p. 21, equation 1.2(38)]). In particular, when

m=p=q=

l,

~ =~=~1=~1=1,

and

z-

1-x

2 '

we shall be led to the main results of Kalla [15] (involving the generalized Jacobi and Rice functions) which, by further specializations, would yield the results given earlier by Blue [2], Gautschi [4], Gatteschi [3], Kalla et al. [5,6], and others. Next, we consider the Brafman polynomials defined by (cf. [10, p. 186]) Bnm[al .... , a p ; ~ l . . . . . ~q : x] = m+pFq [ A ( m ; - n ) ' ° t l ' ' ' ' ' a p ; ] fh . . . . . ~q; z , nENo;

mEN,

(3.2)

Generalized Jacobi and Related Functions

51

where A(m; - n ) abbreviates the array of m parameters

j-n-1 - - ,

j= l,...,m.

m

For p = q = 0, these polynomials would reduce immediately to the generalized Hermite polynomials g'~(x, h) considered by Gould and Hopper [9]. By comparing (3.2) with the definition (1.2), it is easily seen that the Brafman polynomials (3.2) are contained as a special case of Svm(z) with

(at)It" (Olp)k An,k= (131)k" (~q)k' • "

v=n,

z=

and

X

m TM

.

Thus, each of our results can be applied to derive the corresponding integral formulas involving the Brafman and Gould-Hopper polynomials (and, indeed, also their further special cases). For the Bessel polynomials yn(x, c~,/3) defined by (cf. [8, p. 108, equation (34)])

yn(x,a,~)

2Fo[ - n ' a + n - 1 ;

=

~1

.

,

n C Dl0,

and

z = --

(3.3)

our results in the preceding section would apply upon setting

v = n,

m = 1, An,k = (a + n-- 1)k,

X

The details in all these derivations of the special cases are left as an exercise for the interested reader. Finally, we recall the following result:

~-~ (--n)k k=O

_

(~ +

+

+ 1)k (3' + 1)k [~b(3' + k + 1) - ~ ( ~ + 3 ' + k +2)]

n

k! (c~ + 1)k (~ + 3' + 2)k

(/3 + 1)n ( a - 3')n

-

[¢(3' + 1) + ¢ ( a

- 3') - ¢ ( a

- 3' + n)

(3.4)

+

- ¢(/3 +3' + n + 2)],

neNo,

which was derived, on two occasions, by Kalla [5, pp. 155-156, equation (4.4)] and [15, p. 128] as an interesting by-product of his study of integrals involving Jacobi polynomials and Jacobi functions. It may be of interest to observe here that the identity (3.4) can be proven directly by appealing to the familiar Pfaff-Saalschiitz theorem [13, p. 188, equation 4.4(3)]: 3F2[

a,b,-n;] (c-a),~(c-b)n c,a+b-c-n+l; 1 = ~)-~'~--a b)n'

neN0,

(3.5)

which, in turn, would follow readily upon equating the coefficients of z n on both sides of the Euler transformation [13, p. 64, equation 2.1.4(23)]: 2F1 (a, b; c; z) = (1 - z) c-a-b 2F1 (c - a, c - b; c; z) l arg(1 - z)l < l r - e,

(3.6)

0 < ~ < 7r.

Indeed, the first member of the identity (3.4) can easily be rewritten in the form:

r(,~+l)

0,~ I,r(2+3"+2) aF2

c~+1,~+'~+2; 1

which, in vmw of the Pfaff-Saalschiitz theorem (3.5), equals

(~+l)nr(~+~+2) ( a + 1)n

r ( 7 + 1)

o { r(3'+l) c%/

(c~-3'),~ }

FO 3 + 3' + 2) (/3 + 3' + 2)n

which leads us immediately to the second member of the identity (3.4).

'

,

52

R. SRIVASTAVA

REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

H.M. Srivastava, A contour integral involving Fox's H-function, Indmn J. Math. 14, 1-6 (1972). J.L. Blue, A Legendre polynomial integral, Math. Comput. 33, 739-741 (1979). L. Gatteschi, O n some orthogonal polynomial integrals, Math. Comput. 35, 1291-1298 (1980). W. Gautschi, On the preceding paper "A Legendre polynomial integral" by James L. Blue, Math. Comput 33, 742-743 (1979). S.L. Kalla, Some results on Jacobi polynomials, Tamkang J. Math. 15, 149-156 (1984). S.L. Kalla, S. Conde and Y.L. Luke, Integrals of Jacobi functions, Math. Comput. 38, 207-214 (1982). G. SzegS, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23, Fourth edition, Amer. Math. Soc., Providence, RI, (1975). H.L. Krall and O. Frink, A new class of orthogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Sac. 65, 100-115 (1949). H.W. Gould and A.T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J. 29, 51-63 (1962). F. Brafman, Some generating functions for Laguerre and Hermite polynomials, Canad. J. Math. 9, 180-187 (1957). H.M. Srivastava and R. Panda, A note on certain results involving a general class of polynomials, Boll. Un. Mat. Ital. A 16 (5), 467-474 (1979). H.M. Srivastava and N.P. Singh, The integration of certain products of the muitivariable H-function with a general class of polynomials, Rend. Circ. Mat. Palermo 32 (2), 157-187 (1983). A. Erddlyi, W. Magnus, F. Oberhettinger and F.G. 2~ricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, (1953). H.M. Srivastaw and P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, (1985). S.L. Kalla, Integrals of generalized Jacobi functions, Proc. Nat. Acad. Sci. India Sect. A 58, 123-128 (1988).