Integrals associated with generalized k- Mittag-Leffer function

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1 NTMSCI 3, No. 3, 3-5 (5) 3 New Trends in Mathematical Sciences Integrals associated with generalized k- Mittag-Leffer function Chenaram Choudhary and Palu Choudhary Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur. Received: February 5, Revised: 4 March 5, Accepted: 9 June 5 Published online: 4 July 5 Abstract: The aim of the paper is to investigate integrals of Generalized k- Mittag-Leffler function 6, multiplied with Jacobi polynomials, Legendre polynomials, Legendre function, Bessel Maitland function, Hypergeometric function and Generalized hypergeometric function. Keywords: Generalized k- Mittag-Leffler function, Jacobi polynomials, Legendre polynomials, Legendre function, Bessel Maitland function, Hypergeometric function and Generalized hypergeometric function. Introduction The k- Gamma function 5 defined as where (x) n,k is the k- Pochhammer symbol and is given by k n (nk) k x Γ k (x) Lim,k >, x C\kz, () n (x) n,k (x) n,k x(x + k)(x + k)...(x + (n )k), () k, x C\kz,n N +. The integral form of the generalized k- Gamma function 5 is given by Γ k (z) e tk k t z dt, (3) where k R, x C\kz, Re(x) >, from which it follows easily that ( Γ k (γ) k k x γ ) Γ, (4) k and ( (γ) nq,k (k) nq γ ) k nq (5) Let α,β,γ C,k R,{Re(α),Re(β),Re(γ) > } and q (,) N, then the generalized Corresponding author paluchoudhary@gmail.com c 5 BISKA Bilisim Technology

2 4 C. Choudhary and P. Choudhary: Integrals associated with generalized k- Mittag-Leffer function k-mittag-leffler function denoted by k,α,β (z) and defined 6, as k,α,β (z) (γ) nq,k z n n Γ k (αn + β) (6) where (γ) nq,k is the k- Pochhammer symbol given by equation () and Γ k (x) is the k-gamma function given by equation (3). The Generalized Pochhammer symbol (cf, page ), (γ) nq Γ (γ + nq) Γ (γ) q qn q ( ) γ + r i f q N. (7) r q n Integrals with jacobi polynomial The Jacobi polynomial P n (α,β) (x) may be defined by P n (α,β) (x) ( + α) ( n F n, + α + β + n; + α; x ) (8) when α β then the polynomial in (8) becomes the Legendre polynomial (, p. 57). P n (α,β) (x)from (8) it follows that is a polynomial of degree n and that P n (α,β) () ( + α) n (9) Theorem. If α >, β > ; η, µ,γ C, Re(η) >,Re(µ) >,Re(γ) >,k Rand q (,) N. Let is the Jacobi polynomial defined in (8) and Generalized k- Mittag-Leffler function by (6) then we have ()n n+δ+ x λ ( x) α ( + x) δ Pn α,β (x) k,η,µ z( + x)h dx Γ (α + n + ) r Γ (δ+hr+)γ (δ+hr+β+) Γ (δ+hr+β+n+)γ (δ+hr+α+n+) () δ,δ + hr + β +,δ + hr + ; k,η,µ (zh ) 3 F δ + hr + β + n +,δ + hr + α + n + Proof. In dealing with the Jacobi polynomial, we have making use of (6), we get I x λ ( x) α ( + x) δ Pn α,β (x) k,η,µ z( + x)h dx, I x λ ( x) α ( + x) δ Pn α,β (x) r (γ) rq,k z( + x) h r dx. c 5 BISKA Bilisim Technology

3 NTMSCI 3, No. 3, 3-5 (5) / 5 Interchanging order of integration and summation, we can write above expression as I But we have the formula (7 p. 5) r x λ ( x) α ( + x) δ+hr P α,β n (x) dx. () xλ ( x) α ( + x) δ Pn α,β (x)dx ()n α+δ+ λ,δ + β +,δ + ; 3 F, δ + β + n +,δ + α + n + Γ (δ+)γ (α+n+)γ (δ+β+) Γ (δ+β+n+)γ (δ+α+n+) () α >,β >,Re(λ) >. α >,β >,Re(λ) >. provided Now, by using () and () we have This proves result (). I ()n α+δ+hr+ Γ (α + n + ) k,η,µ (zh ) 3 F Γ (δ+hr+)γ (δ+hr+β+) Γ (δ+hr+β+n+)γ (δ+hr+α+n+) λ,δ + hr + β +,δ + hr + ; δ + hr + β + n +,δ + hr + α + n +. Theorem. If (β) >,Re(γ) >,Re(η) >,Re(µ) >,η, µ,γ C,k R, q (,) N. Then we have ( x) δ ( + x) β Pn α,β (x)pm ρ,σ (x) k,η,µ z( x)h dx δ+β+ Γ (α+n+)γ (+ρ+m) m! ( n) r ( m) r (+ρ+σ+m) r (+α+β+n) r r Γ (+ρ+r)γ (α+r+)(r!) k,η,µ (zh )B( + δ + hr + r,β + ). (3) Proof. By Jacobi polynomial, we have I ( x) δ ( + x) β Pn α,β (x)pm ρ,σ (x) k,η,µ z( x)h dx ( x) δ ( + x) β Pn α,β (x)pm ρ,σ (x) r (γ) rq,k z( x) h r dx. Interchanging order of integration and summation, we can write above expression as I r Now, using (8) in above expression we get I r ( + ρ) m m! ( x) δ+hr ( + x) β Pn α,β (x)pm ρ,σ (x) dx. ( m) r ( + ρ + σ + m) + r r ( + ρ) r r ( x) δ+hr ( + x) β Pn α,β (x) dx. (4) r! c 5 BISKA Bilisim Technology

4 6 C. Choudhary and P. Choudhary: Integrals associated with generalized k- Mittag-Leffer function Again using (8) in (4), we have Γ k (ηr+µ)r! I r ( x)δ+hr+r ( + x) β dx. Γ (+ρ+m)γ (+α+n) m! ( n) r ( m) r (+ρ+σ+m) r (+α+β+n) r r Γ (+ρ+r)γ (+α+r) r (r!) (5) But by the formula Then (5) becomes, This proves result (3) ( x) n+α ( + x) n+β dx n+α+β+ B( + α + n, + β + n). (6) I δ+β+ Γ (+ρ+m)γ (+α+n) m! ( n) r ( m) r (+ρ+σ+m) r (+α+β+n) r r Γ (+ρ+r)γ (+α+r) r (r!) k,η,µ (zh ) B( + δ + hr + r, + β). Theorem 3. If Re(α) >,Re(β) >,Re(γ) >,Re(η) >,Re(µ) >,η, µ,γ C,k R and q (,) N. Then the following relation holds true ( x) ρ ( + x) σ Pn α,β (x) k,η,µ z( x)h ( + x) t dx ρ+σ+ ( + α) n ( n) r ( + α + β + n) r r ( + α) r r! k,η,µ (zh+t )B( + ρ + hr + r, + σ +tr). (7) Proof. By invoking Jacobi polynomial and generalized k- Mittag-Leffler function, we have I 3 ( x) ρ ( + x) σ Pn α,β (x) k,η,µ z( x)h ( + x) t dx ( x) ρ ( + x) σ Pn α,β interchanging order of integration and summation, we get I 3 Now, by using (8) in (8), we obtain I 3 r r Making use of (6) in (9), we have This proves Theorem 3. I 3 ρ+σ+ ( + α) n ( + α) n (x) r (γ) rq,k z( x) h ( + x) t r dx, ( x) ρ+hr ( + x) σ+tr P α,β n (x) dx. (8) ( n) r ( + α + β + n) + r r ( + α) r r ( x) ρ+hr+r ( + x) σ+tr dx. (9) r! ( n) r ( + α + β + n) r r ( + α) r r r! k,η,µ (zh+t )B( + ρ + hr + r, + σ +tr). Theorem 4. If Re(α) >,Re(β) >,Re(γ) >,Re(η) >,Re(µ) >,η, µ,γ C,k R and q (,) N. Then the c 5 BISKA Bilisim Technology

5 NTMSCI 3, No. 3, 3-5 (5) / 7 following relation holds true I 4 ( x) ρ ( + x) σ Pn α,β (x) k,η,µ z( x)h ( + x) t dx ρ+σ+ ( + α) n Proof. This Theorem can be proved on similar lines as Theorem 3. ( n) r ( + α + β + n) r r ( + α) r r! k,η,µ (zh t )B( + ρ + hr + r, + σ tr). () Theorem 5. If Re(α) >,Re(β) >,Re(γ) >,Re(η) >,Re(µ) >,η, µ,γ C,k R and q (,) N. Then the following relation holds true Proof. We have ρ+σ+ ( + α) n By using (6), we can write ( x) ρ ( + x) σ Pn α,β (x) k,η,µ z( + x) h dx ( n) r ( + α + β + n) r r ( + α) r r! k,η,µ (z h )B( + ρ + r, + σ hr). () I 5 ( x) ρ ( + x) σ Pn α,β (x) k,η,µ z( + x) h dx. I 5 ( x) ρ ( + x) σ P α,β n Interchanging order of summation and integration, we have Now using (8) in (), we get I 5 r using (6) in (3), we have This proves Theorem 5. r I 5 ρ+σ+ ( + α) n ( + α) n (x) r (γ) rq,k z( + x) h r dx. ( x) ρ ( + x) σ hr P α,β n (x) dx. () ( n) r ( + α + β + n) + r r ( + α) r r ( x) ρ+r ( + x) σ hr dx, (3) r! ( n) r ( + α + β + n) r r ( + α) r r r! k,η,µ (z h )B( + ρ + hr + r, + σ +tr). 3 Special cases (i) If we replace δ by λ- and put α β ρ σ then the integral I transforms into the following integral involving Legendre polynomial, I 6 ( x) λ P n (x) k,η,µ z( x)h dx c 5 BISKA Bilisim Technology

6 8 C. Choudhary and P. Choudhary: Integrals associated with generalized k- Mittag-Leffer function λ r ( n) r ( m) r ( + m) r ( + n) r (r!) (r!) k,η,µ (zh )B(λ + hr + r,). (4) (ii) If α β ρ is replaced by ρ - and σ by σ -, then I 3 transforms into the following integral involving Legendre polynomial, I 7 ( x) ρ ( + x) σ P n (x) k,η,µ z( x)h ( + x) t dx ρ+σ r ( n) r ( + n) r (r!) k,η,µ (zh+t )B(ρ + hr + r,σ +tr). (5) (iii) If αβ, ρ is replaced by ρ- and σ by σ-,, then I 4 transforms into the following integral involving Legendre polynomial, I 8 ( x) ρ ( + x) σ P n (x) k,η,µ z( x)h ( + x) t dx ρ+σ r ( n) r ( + n) r (r!) k,η,µ (zh t )B(ρ + hr + r,σ tr). (6) 4 Integral involving Bessel Maitland function The special case of the Wright function (, vol. 3, section 8.) and (3,) in the form ϕ(b,b;z) ψ ;(B,b);z k Γ (Bk + b) z k k!, (7) with z,b C,B R when B δ, b v + and z is replaced by the function ϕ(δ,v + ;z) is defined by Jv δ known as the Bessel Maitland function (cf 8, p. 35),, which is J δ v (z) ϕ(δ,v + ; z) r ( z) r. (8) Γ (δr + v + ) r! Theorem 6. If η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R, q (,) N,α δα >,α >,Re(ρ + ) >. Then we have the following integral involving Bessel Maitland function, Proof. We have, x ρ J δ v (x) k,η,µ (zxα )dx r interchanging order of summation and integration, we get I 9 Γ (ρ + αr + ) Γ ( + v δ δ(ρ + αr)) GEγ,q I 9 x ρ Jv δ (x) k,η,µ (zxα )dx, x ρ J δ (γ) rq,k (zx v (x) α ) r dx, r r k,η,µ (z). (9) x ρ+αr J δ v (x) dx. (3) c 5 BISKA Bilisim Technology

7 NTMSCI 3, No. 3, 3-5 (5) / 9 By well known formula, ( cf 7,p.55) x ρ J δ v (x)dx where Re(ρ) >, < δ <. We can write (3) as This proves Theorem 6. I 9 r Γ (ρ + ) Γ ( + v δ δρ), (3) Γ (ρ + αr + ) Γ ( + v δ δ(ρ + αr)) GEγ,q k,η,µ (z). 5 Integrals with Legendre functions The Legendre functions are solution of Legendre s differentials equation (9, section 3., vol. ) ( z ) d w dz zdw dz + v(v + ) µ ( z ) w, (3) where z,vµ unrestricted. If we substitute w (z ) µ v in (3) becomes ( z ) d v (µ + )zdv + (v µ)(v + µ + )v, (33) dz dz and if ξ z as the independent variable, this differential equation becomes, ξ ( ξ ) d v dv + ( ξ )(µ + ) + (v µ)(v + µ + )v. (34) dξ dξ This is the Gauss hypergeometric type equation with a µ v, b v + µ + and c µ +. Hence it follows that the function is a solution of (3). w P µ v (z) Γ ( µ) ( ) z + µ F v,v + ; µ; z z, z <, (35) Theorem 7. If η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R, q (,) Nand δ is non negative integer. Then the integral involving Legendre function of first kind written as, xσ ( x ) δ P δ v (x) k,η,µ (zxα )dx ()δ π σ δ Γ (+δ+v) r Γ + (σ+αr) + δ v Γ (σ+αr) Γ + (σ+αr) + δ + v Γ ( δ+v) k,η,µ (z α ). (36) Proof. The integral involving Legendre function of first kind is I x σ ( x ) δ Pv δ (x) k,η,µ (zxα )dx x σ ( x ) δ Pv δ (x) r (γ) rq,k (zx α ) r dx, c 5 BISKA Bilisim Technology

8 C. Choudhary and P. Choudhary: Integrals associated with generalized k- Mittag-Leffer function interchanging order of integration and summation, we can write I r x αr+σ ( x ) δ P δ v (x)dx. Above integral (36)can be solved by using the formula ( 9, section 3., vol. ). xσ ( x ) δ P δ v (x)dx () δ π σ δ Γ (σ)γ (+δ+v) Γ + σ + δ v Γ+ σ + δ + v Γ ( δ+v), (37) where Re(σ) >, δ,,3,... Re(σ) >, δ,,3,... Now, with the help of (3), and (36) can be written as I r () δ π σ αr δ Γ (σ+αr)γ (+δ+v) Γ k (ηr+µ)r! Γ + (σ+αr) + δ v Γ + (σ+αr) + δ + Γ, v ( δ+v) where η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R, q (,) N,Re(σ) >,Re(δ) >. Therefore, This proves result (36). Theorem 8. xσ ( x ) δ P δ v (x) k,η,µ (zxα )dx ()δ π σ δ Γ (+δ+v) r Γ + (σ+αr) + δ v Γ (σ+αr) Γ + (σ+αr) + δ + v Γ ( δ+v) k,η,µ (z α ). Let η, µ,γ C,Re(γ) >,Re(η) >,Re(σ) >,Re(µ) >,k R, q (,) Nand Re(δ) > Then the integral involving Legendre function of first kind is xσ ( x ) δ Pv δ (x) k,η,µ (zxα )dx π δ σ r Γ + (σ+αr) δ v Γ (σ+αr) Γ + (σ+αr) δ v k,η,µ (z α ). (38) Proof. The integral associated with Legendre function of first kind is, I x σ ( x ) δ Pv δ (x) k,η,µ (zxα )dx x σ ( x ) δ P δ (γ) rq,k (zx v (x) α ) r dx, interchanging order of integration and summation, we can write, I r Now, we have the formula ( 9, section 3., vol. ) r x αr+σ ( x ) δ P δ v (x)dx. (39) xσ ( x ) δ P δ v (x)dx π δ σ Γ (σ) Γ + σ δ v Γ+ σ δ v, (4) c 5 BISKA Bilisim Technology

9 NTMSCI 3, No. 3, 3-5 (5) / where Re(σ) >, δ,,3,... Finally, by using (4) in (39), we get I π δ σ r Γ + (σ+αr) δ v Γ (σ + αr) Γ + (σ+αr) δ v k,η,µ (z α ). 6 Integrals with Hermite polynomials H n (x) Hermite polynomials (, p. 87) may be defined as the following relation, valid for all finite and. Since exp(xt t ) H n (x)t n n exp(xt t ) exp(xt)exp( t ) ( )( (x) n t ) n () n t n n n (4) By (4), we can write n H n (x) n H n (x)is a polynomial of degree precisely in x and n / k n / k π n (x)π n (x)where is a polynomial of degree (n-) in x. Theorem 9. the relation holds true, () k (x) n k t n k!(n k)!. () k (x) n k, (4) k!(n k)! H n (x) n x n + π n (x), (43) Let η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R, q (,) N,Re(σ) > and Re(δ) >. Then xρ e x H v (x) k,η,µ (zx h )dx π (v ρ) Γ (ρ hr + ) Γ (ρ hr v + ) GEγ,q k,η,µ (zh ). (44) Proof. The integral associated with Hermite polynomial can be written as, r I x ρ e x H v (x) k,η,µ (zx h )dx x ρ e x H v (x) r (γ)rq,k(zx h ) r dx, c 5 BISKA Bilisim Technology

10 C. Choudhary and P. Choudhary: Integrals associated with generalized k- Mittag-Leffer function interchanging the order of integration and summation, we get Now, by the formula ( 7, p. 59) r x ρ h e x H v (x)dx. (45) Hence, (45) can be written as x ρ e x H v (x)dx π / (v ρ) Γ (ρ + ). Γ (ρ v + ) I π (v ρ) Γ (ρ hr + ) Γ (ρ hr v + ) GEγ,q k,η,µ (zh ). r This proves result (46). Theorem. relation holds true, Let η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R, q (,) N,Re(σ) >,Re(δ) >.Then the I 3 x ρ e x H v (x) k,η,µ (zxh )dx π (v ρ) Proof. This Theorem can be proved on similar lines as Theorem 9. r Γ (ρ + hr + ) Γ (ρ + hr v + ) GEγ,q k,η,µ (z h ). (46) 7 Integrals with hypergeometric function The function F(a,b;c;z) n (a) n (b) n (c) n z n, (47) (a) n Γ (a+n) Γ (a),is known as Hypergeometric function (, p. 45) for c neither zero nor a negative integer, in (47) the notation is the factorial function. Theorem. Let η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R and q (,) N.Then the integral involving hypergeometric function can be written as x ρ (x ) σ F k,η,µ (zx) F v + σ ρ,λ + σ ρ;( x) σ; v + σ ρ,λ + σ ρ; σ; Proof. The integral involving hypergeometric function is, I 4 x ρ (x ) σ v + σ ρ,λ + σ ρ;( x) F σ; r (γ) rq,k x ρ+r (x ) σ F k,η,µ (zx)dx B(r + σ,ρ r σ). (48) k,η,µ (zx)dx, v + σ ρ,λ + σ ρ;( x) σ; dx. c 5 BISKA Bilisim Technology

11 NTMSCI 3, No. 3, 3-5 (5) / 3 Let x t +,then I 4 r (γ) rq,k t σ (t + ) r ρ F v + σ ρ,λ + σ ρ; t σ; dt, r This proves Theorem. (γ) rq,k r k,η,µ (z) F () r (v + σ ρ) r (λ + σ ρ) r (σ) r r! v + σ ρ,λ + σ ρ; σ; t r+σ (t + ) r ρ dt, B(σ + r,ρ r σ). 8 Integrals involving generalized hypergeometric function A generalized hypergeometric function (,p.73) is defined by pf q α,α,...,α p ; z β,β,...,β q ; n where no denominator parameter β j is allowed to be zero or negative integer. Theorem. Let η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R and q (,) N. Then the integral involving generalized hypergeometric function can be written as t p i (α i) n z n q j (β j) n, (49) x ρ (t x) σ p F q (g p );(h q );ax α (t x) β k,η,µ zxu (t x) v dx (5) t σ+ρ f (r)t (α+β)r k,η,µ (ztu+v )B(ρ + ur + αr,σ + vr + βr), r where Re(α),Re(v) o, both are not zero simultaneously. Proof. The integral involving generalized hypergeometric function is Let x st, then r t I 5 x ρ (t x) σ p F q (g p );(h q );ax α (t x) β k,η,µ zxu (t x) v dx, tvr+σ t x ur+ρ ( x t ) vr+σ pf q (g p );(h q );ax α (t x) β dx. I 5 r t (v+u)r tσ+σ s ur+ρ ( s) vr+σ pf q (g p );(h q );as α t α+β ( s) β ds. t σ+σ (γ) rq,k (zt v+u ) r s ur+αr+ρ ( s) vr+βr+σ (g p ) r r r t (α+β)r a r ds, (h q ) r r! c 5 BISKA Bilisim Technology

12 4 C. Choudhary and P. Choudhary: Integrals associated with generalized k- Mittag-Leffer function where f (r) (g p ) r a r r (h q ) r r! (g ) r...(g p ) r a r (h ) r...(h q ) r r! (5) and α, β are non negative integer such that α + β. Theorem 3. Let η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R and q (,) N. Then the integral involving generalized hypergeometric function can be written as t I 6 x ρ (t x) σ p F q (g p );(h q );ax α (t x) β k,η,µ zx u (t x) v dx where f(r) is defined by (5). t σ+ρ f (r)t (α+β)r k,η,µ (zt u v )B(ρ ur + αr,σ vr + βr). (5) r Proof. This theorem can be proved on similar lines as Theorem. Theorem 4. Let η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R and q (,) N. Then the integral involving generalized hypergeometric function can be written as t I 7 x ρ (t x) σ p F q (g p );(h q );ax α (t x) β k,η,µ zxu (t x) v dx where f(r) is defined by (5). t σ+ρ f (r)t (α+β)r k,η,µ (ztu v )B(ρ + ur + αr,σ vr + βr). (53) r Proof. This Theorem can be proved on similar lines as Theorem. Theorem 5. Let η, µ,γ C,Re(γ) >,Re(η) >,Re(µ) >,k R and q (,) N. Then the integral involving generalized hypergeometric function can be written as t I 8 x ρ (t x) σ p F q (g p );(h q );ax α (t x) β k,η,µ zx u (t x) v dx where f(r) is defined by (5). t σ+ρ f (r)t (α+β)r k,η,µ (zt u+v )B(ρ ur + αr,σ + vr + βr). (54) r Proof. This theorem can be proved on similar lines as Theorem. 9 Conclusion In this article we have obtained various integrals involving Generalized k- Mittag-Leffler function. If we set q, then theorems established in this paper reduces for k- Mittag-Leffler function 5. Further if we set k, then the results for the function earlier given by are also obtained. c 5 BISKA Bilisim Technology

13 NTMSCI 3, No. 3, 3-5 (5) / 5 Acknowledgment The Second author would like to thank NATIONAL BOARD OF HIGHER MATHEMATICS, Dept. of Atomic Energy, Govt.of India, Mumbai for the financial assistance under PDF sanction no. /4(47)/3/R&D-II/75. References D.K. Singh and R. Rawat, Integral involving generalized Mittag-Leffler function, J. Frac. Cal. App., 4, 3. E.D. Rainville, Special Functions, New York, Macmillan, E. M. Wright, The asymptotic expansion of the generalized Bessel functions, Proceedings London Mathematical Society, 38, E. M.Wright, The asymptotic expansion of the generalized hypergeometric functions, Journal London Mathematical Society,, G.A. Dorrego and R.A. Cerutti, The k- Mittag-Leffler function, Int. J.Contemp. Math. Sci., 7,. 6 K.S. Gehlot, The Generalized k- Mittag-Leffler function, Int. J. contemp. Math. Sci.,7:45,. 7 V. P. Saxena, The I-Function, Anamaya publisher, New Delhi, 8. 8 V. S. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics, 3, John Wiley and Sons, New York, W. Erdelyi et al, Higher Transcendental Functions,, McGraw Hill, 953. W. Erdelyi, Magnus, F. Oberhettinger, and F. G Tricomi, Higher Transcendental Functions, 3, McGraw- Hill, New York, NY USA, 955. c 5 BISKA Bilisim Technology