ds n SOME APPLICATIONS OF LEGENDRE NUMBERS ps(x) (i x2)s/2dsp (x), KEY WORDS ANY PHRASES. Aoed Legenre functions and polynomials, Legenre polynomials,

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1 Iterat. J. Math. & Math. Sci. VOL. II NO. (1988) SOME APPLICATIONS OF LEGENDRE NUMBERS PAUL W. HAGGARD Departmet of Mathematics, East Carolia Uiversity Greeville, North Carolia 7858 U.S.A. (Received August 6, 1986) ABSTRACT. The associated Legedre fuctios are defied usig the Legedre umbers. From these the associated Legedre polyomials are obtaied ad the derivatives of these polyomials at x are derived by usig properties of the Legedre umbers. These derivatives are the used to expad the associated Legedre polyomials ad x i series of Legedre polyomials. Other applicatios iclude evaluatig certai itegrals, expressig polyomials as liear combiatios of Legedre polyomials, ad expressig liear combiatios of Legedre polyomials as polyomials. betwee Legedre ad Pascal umbers is also give. A coectio KEY WORDS ANY PHRASES. Aoed Legere fuctios ad polyomials, Legere polyomials, divatives of associated Legede polyomi, ad rtegr of Legedre poyo, Legee ad Pasc umbers. 198 MATHAIICS SUBJECT CLASSIFICATION CODES. IOA4, 6C99, 53A INTRODUCTION. The Legedre umbers were itroduced ad may of their elemetary properties were developed i [i]. We apply these pzoperties to a variety of problems ad the use of Legedre umbers may provide somewhat simpler solutios to. DERIVATIVES OF ASSOCIATED LEGENDRE POLYNOMIALS AT x. defied as the problems. For ad s o-egative itegers the associated Legedre fuctios are usual by ps(x) (i x)s/dsp (x), where P (x) is a Legedre polyomial ad D s --- ds pixi s Ox see [i] by usig the Legedre umbers pi, (where Equatio (.1) becomes P (x)= i=o i (.1) Sice P (x) ca be expressed, pi(x) x= ) as (.)

2 46 P.W. HAGGARD ps(x) (I- x)s/d s I i= pixi il ( 3) It is clear that ps(x) is a polyomial of degree for s eve. Thus, let s m ad recall, see [1], that pi for ad i of differet parity. Oittig these ull terms from (.3), oe has pi i (I xz)mdz m 1 x i= (I x )mdm (i) -I pi+l i+i i= Takig the idicated derivatives i (.4) gives (.5) -m (i x)m I x (i+i) pm+ i i i= x (i) eve eve odd. (.4) Pm(x) -m-i pm+iix1+ 1 (.5) (I x) m i= (i+i) To obtai DkFm(x), Leibiz Theorem is used. For < k <, oe has from Dkpm (x) ilo= Dk-i(l_ x)mdii.= [[ki]dk_i(l_x)mdi -m-i i= i= Pm+1+Ix1+l] l i odd. eve, odd. From this equatio oe sees that for eve, Dkp m (x)] for k odd, sice x=o i each term the first factor is for i eve ad the secod is O for i odd. (.6) For odd, Dkpm(X)]x= for k eve, sice i each term the first factor is for i odd ad the secod is for i eve. Oe observes that for eve, ad for odd, Further, Di Di -m pm+ixi I, i odd I if (i) -m-i pm+i+l i+i x i= (i+i) J x= pm+i i>-m ( 7) i eve, i _< m,, i eve O, i>-m pm+i, i odd, i < m (.8)

3 SOME APPLICATIONS OF LEGENDRE ERS 47 D k-i (1 x)mlx= O, k- i odd O, k- i > m (.9) m D k-i (-i) j()xj]x= k i eve, k i < m. j--o The derivative o the right i (.9) ca be expressed usig factorial otatio as (-l)j(j)(j)(k-i)xj-k+ k- i eve, k- i < m, k-i x= j=--- which reduces to the first term k-i (-I) k-i (k- i)! k- i eve, m, - for x. Now (.9) ca be expressed as, k i odd Dk-i(l- x)m]x=o O, k- i > m (.1) k-i m k-i (_i) (k- i)!, k- i eve, < m. [] Usig Equatios (.7), (.8), ad (.1) i (.5) ad the observatios followig (.6), oe sees that, eve, k odd Dkpm(x), odd, k eve (.11) x= k! k (-i) k!m!pm+i [ ad k of the same parity, i=o i, () (m-)! where a term i the series above is if k i is odd or if m < -4k_: Also, recall that pm+i for i > - m, for i odd ad eve, ad for i eve ad odd. Equatio (.11) provides a formula for evaluatig Dm(x) ]x=. Of course, the aswers obtaied by (.11) agree with those obtaied by other methods ad ca be easily verified for small itegers k, m, ad. 3. ASSOCIATED LEGENDRE POLYNOMIALS AND x AS SERIES OF LEGENDRE POLYNOMIALS. It is kow that a associated Legedre polyomial ca be expressed as a series of Legedre polyomials. Equatio (.11) ad a table of Legedre umbers, see Table I, ca be used to provide a formula for the coefficiets i the series. To outlie the method, let pm(x) --- [ AiPi i= Take derivatives to obtai other idetities,. Dkpm(x) AiDiPi(x), k i,, 3,. i= (x). (3. i) I these + i idetities let x. Use (.11) o the left sides of the resultig + 1 idetities ad recall that DkPi(X)]x= pk.,-i see [i]. The right sides ca be

4 48 P.N. IGGARD k simplified by usig Pi ukows ca be solved for the ad Ai s (3.1) gives the desired expasio. The for k > i The system of + 1 idetities i + I values of the Ai s ca be obtaied i the order A, A_l, A_ A, by Dp m (x) p -ipm (x) x--o x=o p-i -i k=-l+l,l<i< (3.) where for i eve ad p-i -i i 1 for i odd.,l<i<, p po pl p p3 p4 p5 p6 p7 p8 Let o lo, I, o 4-- o -- o o ,395 8-W o 48 TABLE i. LEGENDRE NIR4BERS pm 135,135,7,5 Now, x ca be expaded i a series of Legedre polyomials i a similar way. x [ AlPi i= ad proceed as i the derivatio of (3.) to obtai i order A, A-l za-" ua as! p (3.3) Ai i k, k odd -i 1 --i j [IPi+j p.i Ai+j i k, k eve. (3.4)

5 With these values of the Ai s kow expasio SOME APPLICATIONS OF LEGENDRE NUMBERS 49 (3.3) gives the desired expasio, which agrees with the [] (-4k+1) P_k (x) k= k! ()-k 4. SOME INTEGRALS INVOLVING LEGENDRE NUMBERS. I [i], the result ip (X)dx P+l (4. i) IP (x) Pm (X) dx 1 +l for ay positive iteger is give. Here, two other importat itegrals are expressed i terms of Legedre umbers. It is kow, see [], that if m, the More geerally, if m ad are differet o-egative itegers, Raiville, [], gives the result (4.) (-m)(-+l) P(X)Pm(X)dx; (1-x)[p(x)P(X)-Pm(X)P(x)] (4.3) dm(p (x))] With a, b i, ad the results pm for m ad dx Pl= m x=o eve ad P P for m ad odd from [i], Equatio (4.3) becomes > l, Therefore, I m ip pip _p pl ra_ra (X)Pm(X) (m-) (m++l) p1, mlad of the same parity PP m (m ) ((m++l) m ad of differet parity. A third itegral ca be evaluated as show by the followig computatio. With ffl i -k -k.i pm x] x l -dx joxp-k(x)dx JO m! m= J i -k pm xm+ -k [ [.)dx J m! m=o -k -i pm xm+ fxp_k. -k J m! m=o -k pm xm++l] i -k.i m=o -k pm -k m. (m++l) o -k pm (x) v -k dx L m! (m++l) m= Sice the value of this itegral is kow oe ca use this value for the series. dx (4.4) (by.) (4.5)

6 41 P.W. HAGGARD Thus, m -k " P-k m! (m+/l) " k! 3 m= ()-k 5. POLYNOMIALS AS LINEAR COMBINATIONS OF LEGENDRE POLYNOMIALS. (4.6) Sice the Legedre polyomials form a simple set, ay polyomial of a sigle variable ca be expressed as a fiite series of Legedre polyomials. Usig Table, this ca be doe i much the same way ay polyomial ca be expressed i terms of factorials. Cosider the problem of expressig H(x) 5x 3 3x / 4x- 3 i terms of Legedre polyomials. obtai a zero remaider. Thus from which By cotiued subtractio of Legedre polyomials we Detachig coefficiets, H(x) 5 P3(x) 5 Diff. -P (x) Dif. 7Pl(X Diff. -4Po (x) Diff H(x) P3(x) + P (x) 7Pl(x) + 4P O(x), H(x) P3(x P(x + 7Pl(X 4P(x Q6 7 8 Q Q Q Q Q Q Q Q O. (5.1) 1 3 3O I , TABLE. pm

7 SOME APPLICATIONS OF LEGENDRE NUMBERS 411 For a secod method, set 5x 3 3x + 4x 3 =- AP3(x) + BP (x) + CPI(X) + DRo(X). Take the first three derivatives with respect to x, let pm idetities ad use () to obtai the system pm -3 AP3 + BP + CPI + DPo x i each of the four 4= AP 3-6 AP Next, use Table i ad solve for A, B -, C 7, ad listed to obtai (5.1) agai. gp More geerally, if V(x) is a polyomial of degree i x, write V(x) [. AiPi(x) iffio take derivatives, let x i the + 1 obtai the Ai s i the order i to i Sice pm for A i m + odd, v(i) (o) A idetities, as () v () p V (i) () [ A jpj -i j=-i+l p-i -i D -4 i the order ad use (5.) (5.3) p(m)(o) pm to i -l,-,l, the secod equatio of (5.4) ca be expressed as -i IAI+J Pi+j i Pi (5.4) i -l, -,.,,,l, (5.5) Table ca be used to evaluate a fiite series of Legedre polyomials as a polyomial i x. As a example, we evaluate -S(x) P7(x) 4P6 (x) 5P 5(x). Detachig coefficiets, P7(x) i- 16 i sp 5 (x) o o o Sum The, S(x) 7 -x + x x ----x +-x Cosider Table 3, which gives values for Lm. The etries show are itegers, a - -4P 6 (x) LEGENDRE AND PASCAL NU}ERS. result that ca be easily proved. The alterate diagoals have etries of the form

8 41 P.W. HAGGARD +l -i P+i i to (-i) (6.1) readig from upper right to lower left. If p/! is factored from L m L L 1 L L 3 L 4 L 5 L 6 L 7 L O 7O B ,86-4,4 1,87 p m TABLE 3. Lm _m m! each etry o such a diagoal, the remaiig factors are (-1)iC(,i). I otatio, oe has +ip-i+i (-1) ipc (, i) (-i)! Equatio (6.) ca be simplified to (_l)i (-i) i: to. -i -i P+i C(,i), i to, p (6.) (6.3) which shows a coectio betwee Legedre umbers ad Pascal umbers. This result ca be easily proved usig the geeral form of the Legedre umbers pm give i [i]. REFERENCES i. HAGGARD, P.W. O Legedre Numbers, Iteratioal Joural of Mathematics ad Mathematical Scieces, Volume 8, Number, 1985, RAINVILLE, E.D. Special Fuctios, The Macmilla Compay, New York, COPSON, E.T. A Itroductio to the Theory of Fuctios of a Complex Variable, Oxford Uiversity Press, Lodo, RICHARDSON, C.H. A Itroductio to the Calculus of Fiite Differeces, C. Va Nostrad Compay, Ic., New York, 1954.

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