CERTAIN HYPERGEOMETRIC GENERATING RELATIONS USING GOULD S IDENTITY AND THEIR GENERALIZATIONS

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1 Asia Pacific Joual of Mathematics, Vol. 5, No. 08, 9-08 ISSN CERTAIN HYPERGEOMETRIC GENERATING RELATIONS USING GOULD S IDENTITY AND THEIR GENERALIZATIONS M.I.QURESHI, SULAKSHANA BAJAJ, Depatmet of Applied Scieces ad Humaities,Faculty of Egieeig ad Techology, Jamia Millia Islamia A Cetal Uivesity, New Delhi - 005, Idia Depatmet of Applied Scieces ad Humaities, Faculty of Mathematics, Modi Istitute of Techology, Rawatbhata Road, Kota, Rajastha- 400, Idia Coespodig autho: sulakshaa80@ymail.com Received Nov 4, 07 Abstact. I the peset pape, we have obtaied hypegeometic geeatig elatios associated with two hypegeometic polyomials of oe vaiable H,β x; m ad B,β x; m, λ, µ with thei idepedet demostatios via Gould s idetity.as applicatios,some well kow ad ew geeatig elatios ae deduced.usig bouded sequeces, futhe geealizatios of two mai hypegeometic geeatig elatios have also bee give fo two geealized polyomials S,β x; m ad T,β x; m, λ, µ. 00 Mathematics Subject Classificatio.Pimay C05, C5, C0; Secoday C45. Key wods ad phases. Jacobi polyomials; geealized Laguee polyomial; geealized Rice polyomial of Khadeka; Gould s idetity.. Itoductio ad pelimiaies Thoughout i the peset pape, we use the followig stadad otatios: N :,,,..., N 0 : 0,,,,... N 0, Z 0 : 0,,,,..., Z :,,,... Z 0 \0 ad Z Z 0 N. Hee, as usual, Z deotes the set of iteges, R deotes the set of eal umbes, R + deotes the set of positive eal umbes ad C deotes the set of complex umbes. The Pochhamme symbol o the shifted factoial λ ν λ, ν C is defied, i tems of the familia Gamma fuctio, by Γλ + ν ν 0; λ C\0. λ ν : Γλ λλ +... λ + ν N; λ C it is beig udestood covetioally that 0 0 ad assumed tacitly that the Gamma quotiet exists. Some useful cosequeces of Lagage s expasio, p.;see also 5, p.46,poblem 07 c 08 Asia Pacific Joual of Mathematics 9

2 iclude the followig geealizatio 5, p.49,poblem 6 of the familia biomial expasio :. θ + β + t + ζθ+ whee θ+β+ is a biomial coefficiet ad θ, β ae complex umbes idepedet of ad ζ is a fuctio of t defied implicitly by. ζ t + ζ +β subject to the coditio.4 ζ0 0 Aothe geealizatio 5, p.48,poblem elated with the equatio.,is give as: θ θ + β + t θ + β θ θ + β + t whee ζ is defied by the equatios. ad.4. + ζθ Whe β,both esults. ad.5 educe immediately to the biomial expasio. Gould, p.90; see also 7, p.69 gave the followig idetity: θσ + µ θ + β +.6 θ + β + t + ζ θ σ + µθζ βζ whee θ, β, σ, µ ae complex paametes idepedet of ad ζ is give by the equatios. ad.4. If we put θ + β + m ad σ λ + µ m i Gould s idetity.6, we get the fist modified fom of Gould s idetity: + β + mλ + µ + µ m + β + m + β ζ +β+m λ + µ m + with ζ t + ζ β+ ; ζ0 0 + β + m + β + µ + β + mζ t 9

3 If we put θ + β + m ad σ λ + µ β + i Gould s idetity.6, we get the secod modified fom of Gould s idetity: + β + mλ + µ + µ β + + β + m + β ζ +β+m λ + µ β + + with ζ t + ζ β+ ; ζ0 0 + β + m + β + µ + β + mζ t Gauss s Multiplicatio Theoem Fo evey positive itege m, we have m b + j.9 b m m m m j ; 0,,,... Summatio idetity, p.0,lemma,..6.0 m B, B, + m x deotes the geatest itege i x ; m N, povided that seies ivolved ae absolutely coveget. The geealized Laguee polyomials L x 6, p.00,ae defied by. L x + F! ; + ; x ; N 0 Replacig by + β i equatio.,we get ;. F + + β; x! L +β x + + β The Jacobi Polyomials of fist kid P,β x 6, p.54.,p.55.7 ae defied by the followig equatios:. P,β x + F!, + + β + ; + ; x.4 P,β x + + β x! + + β F, ; β ; x 94

4 whee is a o-egative itege. Replacig by + b ad β by β b + i equatio., we get, + + β; x! Γ + b + +b, β b+ F.5 P x + + b ; Γ + b + + Replacig by ad β by β b+ i equatio.4,we get the followig esult, ; Γ + + β b!.6 F P,β b x b β; x Γ + + β b x The geealized Rice Polyomials H,β ν, σ, x of Khadeka 4, p.58,eq.. ae defied by +.7 H,β, + β + +, ν; ν, σ, x F +, σ ;.8 H ν, σ, x H 0,0 ν, σ, x.9 P,β x H,β ν, ν, x Replacig by + b ad β by β b + i eq..7, we get, ν, + + β ;!.0 F x H +b,β b+ ν, σ, x + + b, σ; + + b Some useful Pochhamme s elatios λ+µ λ + µ µβ+ m +. λ + µ + µβ + µ m. λ + µζ + µ m + λ+µ µβ+ m µζβ m + µζ m. + mβ + λ + µ ζ + µ β + + µ ζ β m β ζ λ β ζ + µ ζ β ζ λ βζ+µ ζ λ + µ ζ + µ mζ+ + mβ+ mβ+ β ζ + µ ζ m β ζ λ β ζ+µ ζ + µ β+ β ζ+m ζ λ β ζ+µ ζ µ β+ β ζ+m ζ 95 x λ βζ+µ ζ µ mζ+.a

5 whee 0,,,,... Now we shall discuss some special cases of the implicit fuctios defied by equatio. subject to the coditio.4.usig Mathematica 9.0, we ca fid the oots of esultig cubic equatio i ζ fo diffeet values of β i equatio.. Case I:- Whe β 0 i.,the paticula value of ζsatisfyig the coditio.4 is deoted by.4 Θ t t Case II:- Whe β i.,we get tζ + t ζ + t 0 the oe of the values of ζ satisfyig the coditio.4 is give by.5 Λ t 4t t Case III:- Whe β i.,we get ζ + ζ t 0 the the paticula value of ζsatisfyig the coditio.4 is give by.6 Ξ + + 4t Case IV:- Whe β i., we get.7 Υ + ζ + ζ + ζ t 0 the oe of the ootssatisfyig the coditio.4 of above equatio is give by + 7t + 4t + 7t + 7t + 4t + 7t + Case V:- Whe β i., we get ζ t ζ t 0 the oe of the ootssatisfyig the coditio.4 of above equatio is give by.8 U t t + t

6 Case VI:- Whe β i.,we get ζ + ζ t 0 the oe of the ootssatisfyig the coditio.4 of above equatio is give by Ψ + + ι + 7t + 4t + 7t 4 ι + 7t + 4t + 7t whee ι. Case VII:- Whe β i.,we obtai ζ t ζ t ζ t 0 the oe of the values of ζ satisfyig the coditio.4 is deoted by Π t 6t t 6 7t + 8t 6 + t t 6 + 4t 9.0 Case VIII:- Whe β 7t + 8t 6 + t 9 + 7t 6 + 4t 9 i.,we obtai. ζ t ζ t 0 the oe of the values of ζ satisfyig the coditio.4 is deoted by. Φ Fist Geeatig Relatio: t 9t + 7t 6 4t 9 + 9t + 7t 6 4t 9. Mai geeatig elatios If ay values of vaiables ad paametes leadig to the esults which do ot make sese, ae tacitly excluded, the λ + µ + β + H,β x; m t + ζ. 97

7 λ + µζ.. p+f q+ whee, + λ βζ+µζ, a m β+ m µ +ζ p ; +, b q ; x ζm, λ βζ+µζ m β+ m µ +ζ ζ t + ζ +β ; ζ0 0. povided that ivolved seies o both sides ae absolutely coveget. Hee Sivastava s geealized hypegeometic polyomials of oe vaiable H,β x; m, p.60,eq.7..;see also, pp.- ae give by + β +. H,β m;, a p ; x; m p+mf q+m m; + + β, b q ; x whee,β ae complex paametes idepedet of ad m; λ abbeviates the aay of m umbe of paametes give by λ m, λ + m,..., λ + m ; m N m Idepedet Demostatio: Usig the defiitio. of H,β x; m ad the the powe seies fom of p+m F q+m x i left had side of equatio.,we get λ + µ Ω + β + H,β x; m t λ + µ Γ + β β + Γ + Γ + β +. m. m + j a p x t m j m + + β + j b q! m j Usig Gauss s multiplicatio theoem.9 i above equatio, we get. Ω m λ + µ m β+ a p x t + β + + β m b q!! Now applyig summatio idetity.0 ad the simplifyig futhe, we get mβ+ a p x t m + β + mλ + µ + µ m Ω. + mβ+ b q! + β + m + β + + β + m + β +.4. t Now usig fist modified Gould s idetity.7,we get Ω + ζ mβ+ a p x t m + mβ+ b q! + ζ mβ+. 98

8 .5. λ + µζ + µ m + Now usig. ad simplifyig it futhe, we get λ + µζ.6 Ω + ζ µζβ + m mβ+ a p x ζ m + mβ+ b q!. + µ m+ζ λ βζ+µζ λ βζ+µζ µ m+ζ Afte solvig it futhe,we get the esult. i the fom of a geealized hypegeometic fuctio of oe vaiable. Secod Geeatig Relatio: If ay values of vaiables ad paametes leadig to the esults which do ot make sese, ae tacitly excluded, the λ + µ + β + B,β x; m, λ, µ t + ζ. λ + µζ.7. p+f q+ whee, + m β+ + m β+, ζ t + ζ +β ; ζ0 0. λ βζ+µζ, a µ +β βζ+mζ p; λ βζ+µζ, b µ +β βζ+mζ q; x ζm povided that ivolved seies o both sides ae absolutely coveget. Hee we defie ew geealized hypegeometic polyomialsb,β x; m, λ, µ,kow as Patha s geealized hypegeometic polyomials of oe vaiable, give by.8 B,β x; m, λ, µ + β + p+m+f q+m+ m;, + λ+µ, a µβ+ m p; λ+µ m; + + β,, b µβ+ m q; x whee,β ae complex paametes idepedet of ad m; λ abbeviates the aay of m umbe of paametes give by Idepedet Demostatio: λ m, λ + m,..., λ + m ; m N m Usig the defiitio.8 of B,β x; m, λ, µ ad the the powe seies fom of p+m+ F q+m+ x i left had side of equatio.7,usig Gauss s multiplicatio theoem.9 ad esult.,we get Ω λ + µ + β + B,β x; m, λ, µ t 99

9 .9 m λ + µ + µβ + µ m m Γ + β + a p x t Γ + β + + m b q!! Now applyig summatio idetity.0 i above equatio the simplifyig futhe, we get Ω a p x t m + mβ + b q! + β + m + β +.0. t + β + mλ + µ + µβ +. + β + m + β + Now usig secod modified Gould s idetity.8 ad eq..,we get a Ω + ζ mβ+ p x ζ m + b mβ+ q!.. λ + µ ζ + µβ + + µζβ + m Now usig equatio.a i above equatio ad summig it up ito hypegeometic fom futhe, we get the desied esult.7.. Kow applicatios of fist geeatig elatio. i. Puttig λ, µ β+. i equatio. ad afte simplifyig, we get + ζ+ a p ; pf q b q ; x ζm H,β x; mt which is the esult of Sivastava 9, p.975;see also 0, p.,eq..hee ζ beig give by equatio. ad.4 ad H,β x; m is give by equatio.. ii. Puttig λ, µ β+ x; ad afte simplificatio, we get H,β + β + ad m i equatio. ad usig the defiitio of p+f q+, a p ; + + β, b q ; x t. + ζ+ p F q a p ; b q ; x ζ which is the esult of Sivastava 8, p.59,eq.9;see also, p.86.ζ is give by equatios. ad.4. 00

10 iii. Puttig λ, µ, β x; f H, x ad eplacig ζ by U, we get + f xt p+f q+ ad m i equatio.,usig the defiitio. of, a p ; +, b q ; x t. + U+ + U p F q a p ; b q ; xu whee U t t + t + 4 which is the kow esult of Bow, p.64,eq.7 ad f x is Bow s geealized hypegeometic polyomial, p.58,eq New applicatios of fist geeatig elatio. i. Puttig β 0 ad ζ Θ λ + µ + + t t p+mf q+m fom equatio.4 i equatio., we get m;, a p ; m; +, b q ; x t t. λ t + µ t 4.. p+f, + λ t+µ t, a m m µ p ; q+ +, λ t+µ t, b m m µ q ; x m t t ii. Puttig β ad ζ Λ t 4t t + p+mf q+m λ + µ + fom equatio.5 i equatio.,we get m;, a p ; m; + +, b q ; x t + Λ. λ Λ + µ Λ 4.. Λ p+f q+, + λ Λ+µ Λ, a m m µ +Λ p ; +, b q ; x Λm, λ Λ+µ Λ m m µ +Λ iii. Puttig β ad ζ Ξ + +4t fom equatio.6 i equatio.,we get λ + µ m;, a p ; p+mf q+m m; +, b q ; x t + Ξ. λ + Ξ + µξ Ξ p+f q+ m, + λ +Ξ+µΞ m µ +Ξ, a p ; m, λ +Ξ+µΞ m µ +Ξ, b q ; x Ξ m 0

11 iv. Puttig β ad ζ Υ fom equatio.7 i equatio.,we get λ + µ m;, a p ; p+mf q+m m; +, b q ; x t + Υ. λ + Υ + µυ Υ p+f, + λ+υ+µυ, a m m µ +Υ p ; q+, b q ; x Υm, λ+υ+µυ m m µ +Υ v. Puttig β λ + µ + ad ζ U fom equatio.8 i equatio.,we get + m;, a p ; p+mf q+m m; +, b q; x t + U. λ + U µ U + U p+f q+ U +µ U, + λ+, a m m µ +U p ; +, λ+ U +µ U m m µ +U, b q ; x Um vi. Puttig β λ + µ ad ζ Ψ fom equatio.9 i equatio.,we get m;, a p ; p+mf q+m m; +, b q; x t + Ψ. λ + Ψ + µψ Ψ p+f q+ m, + λ+ Ψ+µΨ mµ+ Ψ, a p ; m, λ+ Ψ+µΨ mµ+ Ψ, b q ; x Ψ m vii. Puttig β λ + µ + ad ζ Π fom equatio.0 i equatio.,we get + m;, a p ; p+mf q+m m; +, b q; x t + Π. λ + Π + µπ Π p+f q+, + λ+ Π+µΠ, a m mµ+ Π p ; +, b q ; x Πm, λ+ Π+µΠ m mµ+π viii. Puttig β λ + µ + ad ζ Φ fom equatio. i equatio.,we get + m;, a p ; p+mf q+m m; +, b q; x t + Φ. λ + Φ + µφ Φ p+f q+, + λ+ Φ+µΦ, a m m µ +Φ p ; +, λ+ Φ+µΦ, b m m µ +Φ q ; x Φm 0

12 5. New applicatios of secod geeatig elatio.7 i. Puttig λ, µ β+ + β + 5. i equatio.7 ad afte simplifyig, we get p+m+f q+m+ + ζ+ p+f q+ m;, + +β+, a β+β+ m p; +β+ m; + + β,, b β+β+ m q; x, + mβ+ + mβ+, whee ζ is give by equatios. ad.4. +ζ, a β+ βζ+mζ p; +ζ, b β+ βζ+mζ q; x ζm ii. Puttig λ, µ β+, m i equatio.7 ad usig the defiitio of B,β x;,, β+, we get + β +, + p+f +β+, a p; q ζ+ p+f q+ + + β,, + β+ + β+, whee ζ is give by equatios. ad.4. +ζ β+β +β+ β+β, b q; x, a β+ βζ+ζ p; +ζ, b β+ βζ+ζ q; x ζ iii. Puttig λ, µ, β, m i equatio.7,usig the defiitio of B, x;,, ad eplacig ζ by U fom equatio.8, we get +, 4, a p ; p+f q+ +, 4, b q ; x t 5. + U+, + 8+U + U p+f, a +U p; q+ +, 8+U, b +U q; x U iv. Puttig β 0 ad ζ Θ t fom equatio.4 i equatio.7, we get t λ + µ + m;, + λ+µ p+m+f, a µ m p; q+m+ + λ+µ m; +,, b µ m q ; x t t. λ t + µ t 5.4. p+f, + λ t+µ t, a m µ t+mt p ; m t q+ + λ t+µ t,, b m µ t+mt q ; x t v. Puttig β ad ζ Λ t 4t fom equatio.5 i equatio.7,we get t λ + µ + m;, + λ+µ p+m+f, a µ m p; q+m+ + λ+µ m; + +,, b µ m q; x t t t 0

13 λ Λ + µ Λ Λ Λ p+f q+ λ Λ+µ Λ, + m + m, a µ Λ+mΛ p; λ Λ+µ Λ,, b µ Λ+mΛ q; x Λm vi. Puttig β ad ζ Ξ + +4t fom equatio.6 i equatio.7,we get λ+µ λ + µ m;, p+m+f, a µ+m p ; q+m+ λ+µ m; +,, b µ+m q; x t λ + Ξ + µξ Ξ + Ξ λ +Ξ+µΞ,, a m µ +Ξ+mΞ p ; p+f q+ λ +Ξ+µΞ,, b m µ +Ξ+mΞ q; x Ξm vii. Puttig β ad ζ Υ fom equatio.7 i equatio.7,we get λ+µ λ + µ m;, p+m+f, a µ+m p; q+m+ λ+µ m; +,, b µ+m q; x λ + Υ + µυ Υ + Υ viii. Puttig β λ + µ + p+f q+, λ+υ+µυ, a m µ +Υ+mΥ p;, λ+υ+µυ, b m µ +Υ+mΥ q; x Υm ad ζ U fom equatio.8 i equatio.7,we get + λ+µ m;, + p+m+f q+m+ µ m, a p; m; + λ+µ, µ m, b q; x t t λ + U U + µ U + U p+f q+, + λ+ U +µ U, a m µ + U +mu p ; +, λ+ U +µ U, b m µ + U +mu q ; x Um ix. Puttig β λ + µ ad ζ Ψ fom equatio.9 i equatio.7,we get λ+µ m;, p+m+f q+m+ µ +m, a p; m; + λ+µ, µ +m, b q; x t λ Ψ Ψ + µψ + Ψ p+f q+, λ+ Ψ+µΨ, a m µ+ p; Ψ+m Ψ, λ+ Ψ+µΨ, b m µ+ Ψ+m Ψ q ; x Ψm x. Puttig β λ + µ + ad ζ Π fom equatio.0 i equatio.7,we get + λ+µ m;, + p+m+f q+m+ µ m, a p; m; + λ+µ, µ m, b q; x t 04

14 λ Π Π + µπ + Π xi. Puttig β λ + µ + p+f q+, + λ+ Π+µΠ, a m µ+ p; Π+mΠ +, λ+ Π+µΠ, b m µ+ q; x Πm Π+mΠ ad ζ Φ fom equatio. i equatio.7,we get + λ+µ m;, + p+m+f q+m+ µ m, a p; m; + λ+µ, µ m, b q; x t λ Φ Φ + µφ + Φ p+f q+, + λ+ Φ+µΦ, a m µ + Φ+mΦ p ; +, λ+ Φ+µΦ, b m µ + Φ+mΦ q ; x Φm Makig suitable adjustmets of paametes ad vaiables i all geeatig elatios of sectios 4 ad5, we ca also obtai a umbe of ew geeatig elatios ivolvig esticted geealized Laguee polyomials, esticted Jacobi polyomials, esticted geealized Rice polyomials of Khadeka ad othe othogoal polyomials. Let 6. Futhe geealizatios of geeatig elatios. ad.7 Geealizatio of.: m 6. S,β x; m + β + m γ x whee,β ae complex paametes idepedet of ; m is a abitay positive itege ad γ is a bouded sequece of abitay eal ad complex umbes such that γ 0.The λ + µ + β + S,β x; mt 6. whee ζ is give by + ζ λ + µζ + µ m + ζ γ x ζ m + β + m 6. ζ t + ζ β+ ; ζ0 0 povided that each of the seies ivolved is absolutely coveget. Idepedet Demostatio: Usig the defiitio 6. of S,β x; m i left had side of equatio 6.,we get Ω λ + µ + β + S,β x; m t 05

15 6.4 m λ + µ Γ + β β + Γ m + Γ + β + m + γ x t Applyig summatio idetity.0 ad the simplifyig futhe, we get Ω γ x t m + β + mλ + µ + µ m + mβ + + β + m + β β + m + β + t Now usig fist modified Gould s idetity.7 with coditio 6.,we get 6.6 Ω + ζ λ + µζ µζβ + m γ x ζ m + µ m + + mβ + Chagig the summatio idex fom to ad afte solvig it futhe,we get the geeal esult 6. coespodig to ou fist geeatig elatio. subject to the coditios 6.. Geealizatio of.7: Let 6.7 T,β x; m, λ, µ m λ + µ + β + λ + µ + µβ + m γ x m whee, β,λ,µ ae complex paametes idepedet of ; m is a abitay positive itege ad γ is a bouded sequece of abitay eal ad complex umbes such that γ 0.The λ + µ + β + T,β x; m, λ, µt ζ whee ζ is give by λ + µζ + µ + β βζ + mζ γ x ζ m + β + m ζ t + ζ β+ ; ζ0 0 povided that each of the seies ivolved is absolutely coveget. Idepedet Demostatio: Usig the defiitio 6.7 of T,β x; m, λ, µ i left had side of equatio 6.8,we get Ω λ + µ + β + T,β x; m, λ, µ t 06

16 6.9 m + β + λ + µ + µβ + µ m m + β + γ x t Applyig summatio idetity.0 ad the simplifyig futhe, we get Ω γ x t m + β + mλ + µ + µ β + + mβ + + β + m + β β + m + β + t Now usig secod modified Gould s idetity.8 with coditio 6.,we get 6. Ω + ζ λ + µζ + µ β + + µζβ + m γ x ζ m + mβ + Chagig the summatio idex fom to ad afte solvig it futhe,we get the geeal esult 6.8 coespodig to ou secod geeatig elatio.7 subject to the coditios 6.. I the defiitios of geealized polyomials give by S,β x; m ad T,β x; m, λ, µ, puttig γ m a... a p, b... b q! we obtai Sivastava s geealized hypegeometic polyomials of oe vaiable H,β x; m ad Patha s geealized hypegeometic polyomials of oe vaiable B,β x; m, λ, µ espectively. Refeeces Bow, J.W.; New Geeatig Fuctios fo Classical Polyomials, Poc. Ame. Math. Soc., 969, Gould, H.W.; Some geealizatios of Vademode s Covolutio, Ame. Math. Mothly, 6956, Kaade, B.K. ad Thakae, N.K.; Note o the geeatig fuctio fo geealized hypegeometic polyomials, Idia J. Pue Appl. Math., 6975, Khadeka, P.R.; O a geealizatio of Rice s polyomial, I. Poc. Nat. Acad. Sci. Idia Sect. A, 4964, Pólya, G. ad Szegö, G; Poblems ad Theoems i Aalysis, Vol.I, Taslated fom the Gema by D.AeppliSpige-Velag, New Yok, Heidelbeg ad Beli Raiville, E.D.; Special Fuctios, The Macmilla, New Yok, 960; Repited by Chelsea Publishig Compay, Box, New Yok, Rioda, J.; Combiatoial Idetities, Joh Wiley & Sos, New Yok, Lodo ad Sydey

17 8 Sivastava, H.M.; Geeatig Fuctios fo Jacobi ad Laguee Polyomials, Poc. Ame. Math. Soc., 969, Sivastava, H.M.; A class of Geeatig Fuctios fo geealized Hypegeometic PolyomialsAbstact, Notices Ame. Math. Soc., 6969, 975Abstact #69T B98. 0 Sivastava, H.M.; A class of Geeatig Fuctios fo geealized Hypegeometic Polyomials, J. Math. Aal. Appl., 5 97, 0 5. Sivastava, H.M.; A Note o Cetai Geeatig fuctios fo the Classical Polyomials, Atti Accad. Naz. Licei Red. Cl. Sci. Fis. Mat. Natu.8, 6977, 8. Sivastava, H. M. ad Maocha, H. L.; A Teatise o Geeatig fuctios, Halsted Pess Ellis Howood Ltd., Chicheste, U.K., Joh Wiley ad Sos, New Yok, Chicheste, Bisbae ad Tooto, 984. Whittake, E.T.ad Watso, G.N.; A Couse of Mode Aalysis, Fouth ed., Cambidge Uiv.Pess, Cambidge, Lodo ad New Yok