ON SOME SPECIAL POLYNOMIALS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 4, Pages S ) Article electronically published on November 25, 2003 ON SOME SPECIAL POLYNOMIALS V V KARACHIK Communicated by Carmen C Chicone) Abstract New special functions called G-functions are introduced Connections of G-functions with the nown Legendre, Chebyshev and Gegenbauer polynomials are given For G-functions the Rodrigues formula is obtained Introduction Let L, L 2 be linear partial differential operators acting on functions that belong to the vector space X such that L X X =, 2) and defined in some domain Ω R n DenoteÑ = N {0} Definition [2]) An infinite ordered system of functions {f x) Ñ} f X) is called f-normalized with respect to L,L 2 ) in the domain Ω, having a base f 0 x) if everywhere in that domain, L f 0 x) =fx) andl f x) =L 2 f x) for N The main property of a system of functions f-normalized with respect to operators L,L 2 ) in the domain Ω is the following: the series ux) = =0 f x) satisfies formally the equation L ux) L 2 ux) =fx) inωandthereforeux) can be considered as a formal solution to that equation Example Let n =2,L = y / x, L 2 = / y and Ω = R 2 /{y =0} Then a system of functions 0-normalized with respect to these operators in Ω can be given in the form f x, y) = m m m y m 2 x for =0,, and 2 m N Using the main property of f-normalized systems we can easily construct formal solutions to the equation yu x x, y) u y x, y) = 0 in Ω In this case formal solutions m/2 +) u m x, y) = 2x) y m 2, m N ) =0 for some x, y 2x <y 2 ) are real solutions because the corresponding series are convergent at these points For even m we get polynomial solutions A very important particular case of the above notion is the case when L 2 = I, where I is the identity operator Then the system of functions {f x) Ñ} Received by the editors November 5, Mathematics Subject Classification Primary 33D45; Secondary 3B05, 35C05, 33D50 Key words and phrases Harmonic polynomials, orthogonal polynomials 049 c 2003 American Mathematical Society

2 050 V V KARACHIK f-normalized with respect to L,I) is called f-normalized with respect to L Elements of the system {f x) Ñ} in this case satisfy in Ω the conditions ) L f 0 x) =fx); L f x) =f x), N { } Example 2 Let L D) = Thesystem h s x) Ñ,where h s x 2 x) = H s x), x 2 = x x 2 2, 2) n +2s, 2) n, a, b) = aa + b) a + b b), with the convention a, b) 0 =herea, ) = a) ;see[5]),andh s x) is a homogeneous harmonic polynomial of order s, is0- normalized with respect to in R n It follows from the equality H s x) x m )= m m + n 2)H s x)+2 n i= x id xi H s x)) x m 2 Indeed, using homogeneity of H s x) we can rewrite the above equality in the form H s x) x m )=mm +2s + n 2)H s x) x m 2 Ifwetaeherem =2, we get x 2 ) 2n +2s +2 2) H sx) = H s x) x 2 ) Dividing both sides by 2, 2) n +2s, 2),weget)forL =,fx) =0 and f x) =h s x), ie, hs x) =hs x) for>0and hs 0x) H s x) =0 { } A very important detail here is that the base of the system h s x) Ñ is an arbitrary homogeneous harmonic polynomial H s x) In a straightforward way from the definition we can establish that the system {L 2f x) Ñ} is f-normalized with respect to L,L 2 ) in domain Ω if, L L 2 f x) =L 2 L f x) in Ω Therefore the series 2) ux) = L 2 f x) is a formal solution to the equation LD)u L ux) L 2 ux) =fx) inω =0 Example 3 Let LD) = Using the above construction we are going to find a relation between harmonic polynomials of n variables and harmonic polynomials of n variables There are two possibilities that use 2) I Let us choose L D) = 2 / x 2 n andl 2 D) = 2 / x 2 n Touse2) we have to find a system of polynomials 0-normalized with respect to We can tae it from Example 2 The system {h s x n ))x m,! n Ñ}, wherex n ) = x,,x n ),t m,! = t m /m! could be such a system Therefore, using 2) we can get a harmonic polynomial of n variables in the form ux) = [m/2] ) x n ) 2 x m 2,! n H s x n ) ) 2, 2) n +2s, 2) =0 If we denote [/2] 3) G s x n) )= ) i x n ) 2i x 2i,! n, 2, 2) i n +2s, 2) i

3 ON SOME SPECIAL POLYNOMIALS 05 then we can assert that multiplication of a homogeneous harmonic polynomial H s x n ) )ofn variables on the polynomial G s x n)) gives a harmonic polynomial of n variables, ux) =G s x n))h s x n ) ) It is proved [2] that any homogeneous harmonic polynomial of n variables can be represented in this way II Let us now choose L D) = 2 / x 2 n and L 2 D) = Then the system { ) H 0 x)x 2,! n + H x)x 2+,! n ) },whereh 0 x) andh x) are arbitrary polynomials in x =x,,x n ) is 0-normalized with respect to the operator L 2 in R n Using 2) we obtain harmonic polynomials in the nown form [6] ux) = ) x 2,! H 0 x)+ ) x 2+,! H x) =0 n =0 n 2 G-polynomials According to Example 3 the following definition is useful Definition A polynomial G s x n)) of the form 3) is called a G-polynomial of degree, orders, and ind n If n = 2 there are only two linearly independent homogeneous harmonic polynomials of degree >0) and according to our construction Example 3, case I) we can write them in the following form: [ s)/2] 4) H s x 2)) =G s s x 2))x s = ) j x 2j+s,! x 2j s,! 2, s =0, j=0 for N, and with the convention H 0 0 It is necessary to note that H0 x 2)) = Rex 2 + ix ),! and H x 2)) =Imx 2 + ix ),! This approach allows us to construct harmonic polynomials as a product of G- polynomials in the form 5) G ν) x n) )=G ν2 ν ν 2 x n) ) G νn ν n ν n x 2) )x νn, where ν Ñn, ν ν n,andν n =0, In[3]itisprovedthatpolynomialsG ν) mae up a basis among homogeneous harmonic polynomials of degree ν and they are orthogonal in L 2 S n ), where S n is the unit ball in R n Moreover, if we denote by P the set of all polynomials over C, with the scalar product P x),qx) = P D)Qx) x=0 see []), then polynomials 5) are orthogonal in P too We need also two results from [3] Lemma G-polynomials of the same order s and ind n are orthogonal with weight ρ s n x) = x2 n )s on S n Lemma 2 Let f C S n ) be taen in the form fx) =ϕ x,x n )P x), where P x) is a homogeneous polynomial of degree and ϕ C S n )Then 6) fx) dx = ϕ x,x n ) x 2 ) /2 n dx P x) d x x = ω n x = x =

4 052 V V KARACHIK 3 Connection of G-functions with Legendre and Chebyshev polynomials Let us consider the trace of a G-polynomial on the unit sphere In this case a new notion of G-function can be presented Definition The function 7) G s,n [ s)/2] t) = ) i t s 2i,! t 2 ) i+s/2 2, 2) i n +2s, 2) i is called the G-function of degree, orders, and ind n It is not difficult to derive from 5) that n G ν) x) = x ν i= G νi+,n i+ ν i cos ϕ n i+ ), where ϕ i = arccosx i / x i) ), ν ν n,andν n =0, Theorem For a G-function of odd ind, the equality G s,2m+3 2s + m)!! t) = + s +2m)! t2 ) m/2 P s+m +m t) holds, where m 0, s 0, andp s t) is the associated Legendre function [4] Proof First let us prove that [m/2] 8) Dt m = m! y) m 2,! 2D y ) m 2, 2) =0 y=t 2 For this purpose assume that for some m the equality 9) yd y ) m = [m/2] =0 a m y) m 2,! D m y holds Of course for m = the above equality is true and a 0 = Applying the operator yd y to the right-hand side of 9) for even m and assuming that a m = 0, we can obtain [m/2] =0 ) 2 am +m 2 +)am y) m 2+,! Dy m + Let us consider the case when m is odd Then the right-hand side of 9) after applying the operator yd y taes the form [m/2]+ =0 Because the function ) 2 am +m 2 +)a m y) m 2+,! Dy m + γm) = { [m/2], [m/2] +, m even m odd

5 ON SOME SPECIAL POLYNOMIALS 053 Figure canbewrittenintheformγm) =[m +)/2], the results are the same for even and odd m Therefore for any m, yd y ) m+ = [m+)/2] =0 ) 2 am +m 2 +)a m y) m 2+,! Dy m + Therefore, for 9) to hold for m = m +, the coefficients a m should satisfy the following recurrence relation: a m+ = 2 am +m 2 +)am for =0,,,[m +)/2], where a m =0anda 0 = In Figure the obtained recurrence relation is illustrated The incoming arrows to the point, m) show the dependencies of the value a m on the values of am and a m if they are nonzero We do not need to consider a m on the line m =2 because the coefficient for it on this line is m 2 +, ie, zero It is clear that because of a m =0wehaveam+ 0 =m +)a m 0 =m +)!a 0 =m + )! Therefore a m is found on the m-axis Now we can find am for =andm 2 In general, the coefficients a m on the line = const are fully determined by the am on the line = const Therefore the solution of the above recurrence relation exists and is unique It is not so important here to show how to find this solution We just tae it in the form a m = m!/4!) Let us chec it: [ ] 2 am +m 2 +)am = m! m 2 +) 4 + )! 2 4 = m +)! 4! = a m+ Substituting the obtained value of a m into 9) and taing into account that D t = 2 yd y we get 8)

6 054 V V KARACHIK Now we can apply the operator equality 8) for m = m + s to the function t 2 ) m This yields 0) Dt m+s t 2 ) m = )m m + s)!2, 2) m 2, 2) s [m s)/2] =0 ) t m s 2,! t 2 ) 2, 2) 2 + 2s, 2) From here, using equalities P st) = t2 ) s/2 Dt sp t), P t) =2 Dt t2 ),!, which give us P s ) t) = 2! t2 ) s/2 Dt +s t 2 ),weobtain [ s)/2] G s,3 t) ) i t s 2i,! t 2 ) i+s/2 = 2s!! 2, 2) i 2 + 2s, 2) i + s)! P s t) Taing into account that G s,2m+3 t) = t 2 ) m/2 G s+m,3 +m t) we complete the proof Consider by analogy with the associated Legendre functions, the associated Chebyshev functions T st) t2 ) s/2 Dt st t), where 0 <s, T 0t) =T t) being the Chebyshev polynomials Theorem 2 For G-functions of even ind the following equality holds: G s,2m+2 2s +2m )!! t) = + s +2m )! + m) t2 ) m/2 T s+m +m t), where m 0, s>0, andg 0,2 t) =/!T 0t) Proof First we prove that for s>0 the following equality holds: ) [/2] yd y ) s ) i y) 2i,! y) i 2, 2) i, 2) i = 2 s [ s)/2] + s )! )!2s )!! ) i y) s 2i,! y) i 2, 2) i + 2s, 2) i Using the property of γ) functions again see proof of 9)) for s 0, we can get [ s)/2] yd y ) ) i y) s 2i,! = 2 [ s )/2] y) i 2, 2) i + 2s, 2) i ) i + s 2i ) y) s 2i,! y) i +2s +2i 2, 2) i + 2s, 2) i = + s [ s )/2] 2 + 2s) ) i y) s 2i,! y) i 2, 2) i + 2s +), 2) i This implies that ) is true for all s Taing y = t 2,whichmeansD t = 2 yd y, we can write ) in the form D s t G0,2 t) = + s )! )!2s )!! t 2 ) s/2 G s,2 t)

7 ON SOME SPECIAL POLYNOMIALS 055 It is not difficult to see that G 0,2 t) =! cos arccos t) =! T t) Therefore t) can be written in the form G s,2 2) G s,2 t) = )!2s )!! + s )! t 2 ) s/2 D s! t T t) = 2s )!! + s )! T s t) To complete the proof we note G s,2m+2 t) = t 2 ) m/2 G s+m,2 +m t) Remar Using 8) we can obtain T t) = ) ) t 2 /2 ) D 2 )!! t t 2 /2 Theorem 3 For G-functions the Rodrigues formula see [4]) 3) ) Dt t 2 ) +s+n 3)/2 =!n +2s, 2) t 2 ) s+n 3)/2 G s,n +s t) holds, and the following connection of G-functions with Gegenbauer polynomials is valid: 4) G 0,n n 3)! t) = n 3+)! Cn/2 [t], n 3 Proof By analogy with 0) and with the help of 8) we can write [/2] Dt t2 ) +s+n 3)/2 =! t 2i,! 2 D i t 2 t 2 ) +s+n 3)/2 2, 2) i = )!n +2s, 2) t 2 ) s+n 3)/2 G s,n +s t), which proves 3) To prove 4) we use the following representation of Gegenbauer polynomials [5] [n/2] 5) Cn ν t) = 2ν) n t n 2i t 2 ) i 2 2i i!ν +/2) i n 2i)!, which leads to [n/2] Cnt) ν =2ν) n ) i t n 2i,! t 2 ) i =2ν) n G 0,2ν+2 n t), 2, 2) i + 2ν, 2) i and then G 0,n t) = n 2) C n/2 t) = n 3)! n 3+)! Cn/2 t) forn>2 Proofis completed 4 Orthogonality of G-functions Theorem 4 G-functions of ind n n 2) and the same order are orthogonal on [, ] with weight ρt) = t 2 ) n 3)/2,and G s,n t))2 t 2 ) n 3)/2 dt = 2 2s+n 3 Γ 2 s +n )/2) +n 2)/2) + s + n 3)! s)! Proof Using Lemma and because of equality G s x) =Gs x,x n)weobtain 0= G s x)g s mx) x 2 n) s dx = ω n G s,n +s t)gs,n m+st)ρt) dt x =

8 056 V V KARACHIK This implies orthogonality of G-functions G s,n From Theorem for odd n 3weget +s t) andgs,n m+s t) withweightρt) G s,n 2s + n 3)!! t) = + s + n 3)! t2 ) n 3)/4 P s+n 3)/2 +n 3)/2 t), and from Theorem 2 for even n 3, we have 6) G s,n t) = 2s + n 3)!! +n 2)/2) + s + n 3)! t2 ) n 2)/4 T s+n 2)/2 +n 2)/2 t) It is nown that P s 2 + s)! t))2 dt = and therefore for odd n 3, 2 + s)! we have 7) G s,n 2s + n 3)!!) 2 t))2 ρt) dt = +n 2)/2) + s + n 3)! s)! Let n 2 be even Since in this case G s,n t) are connected with Chebyshev polynomials, we need to investigate T st) According to the definitions of T st) and G s,n t) and with the help of 2) we get T s + s )! t) = t 2 ) [ s)/2] s/2 ) i t s 2i,! t 2 ) i 2s )!! 2, 2) i n +2s, 2) i If we use representation 5) see [5]) we obtain T st) =2s 2)!! t2 ) s/2 C s s [t] It is nown [5] that for Gegenbauer polynomials Cs [t])2 t 2 ) s /2 dt = 2s) Γ 2 )Γs + 2 )/! + s)γs)) So, we can get 8) T s t)) 2 t 2 ) /2 dt = 2 2s 2)!!) 2 s + )! π s)!2s 2)!!) 2 2 Therefore, whence using 6) we can obtain T s t)) 2 t 2 ) /2 dt = + s )! π s)! 2, 9) G s,n 2s + n 3)!!) 2 π t))2 ρt) dt = +n 2)/2) + s + n 3)! s)! 2 Combining 7) and 9) and using properties of the Γ-function, we get the necessary equality The proof is completed Remar Associated Chebyshev functions of the same order are orthogonal on the segment [, ] with weight ρt) = t 2 ) /2, and the formula 8) holds Polynomials G ν) x) forn =2, 3 turn into well-nown spherical functions [] Theorem 5 The scalar product of G ν) x) on G µ) x) in L 2 S) is equal to the product of scalar products Gν νi+,n i+ i t) on G µ µi+,n i+ i t) in L 2, ) with weights ρ i t) = t 2 ) n 2 i)/2 for i n Proof The proof can be easily obtained by using definition 5) of G ν) x), applying equality 6) and then using the property of H sx 2)) from4)

9 ON SOME SPECIAL POLYNOMIALS 057 Theorem 6 The G-function Ḡs,n t) normalized in L 2, ) with weight ρt) = t 2 ) n 3)/2 satisfies the estimation 20) Ḡs,n t) 2 +n 2)/2) 2 n 3, s Proof To evaluate G s,n t) we need to rewrite it in a different form Using the nown connection of Γ- and B-functions, Bp, s) = Γp)Γs)/Γp + s), and the equality Γp +)=pγp) we can get = α /2 α) s+n 4)/2 dα n +2s, 2) 2 )!! B/2,s+n 2)/2) 0 If s 0andn 2, then from the definition of G s,n t) 7) we obtain B/2,s+n 2)/2) 2 G s,n t) = t 2 ) s/2 + 0 t + i ) s,! α t 2 ) t i α t 2 ) ) s,! ) s+n 4)/2 dα α), α and therefore G s,n t) / s)! From Theorem 4 we derive G s,n t) 2 + n 2 ) + s + n 3)! s)! 2 2s + n 3)!!) 2 Thus, taing into account that 2s + n 3)! 2s + n 3)!!) 2 we get Ḡs,n t) 2 + n 2 ) ) + s + n 3 +n 2)/2))2 2+n This implies that 20) is true If s =0andn =2,then G 0,2 t) =! T t) /! and therefore, according to 8), Ḡ0,2 t) 2/π < Because < 2 /2, the Theorem is true for this case too Acnowledgment Research for this paper was supported in part by the Junior Faculty Development Program, which is funded by the Bureau of Educational and Cultural Affairs of the Department of State, under authority of the Fulbright-Hays Act of 96 as amended, and administrated by the American Council for International Education: ACTR/ACCELS The opinions expressed herein are the author s own and do not necessarily express the views of either ECA or the American Councils I would lie also to than Professor Paul E Sacs for his valuable remars References [] E M Stein and G Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, NJ, 97 MR 46:402 [2] V V Karachi, Polynomial solutions to systems of partial differential equations with constant coefficients, Yoohama Mathematical Journal 47 2) 2000), 2 42 MR 200d:3503 [3] V V Karachi, On one set of orthogonal harmonic polynomials, Proc Amer Math Soc 26 2) 998), MR 99g:33034 [4] A Erdélyi,WMagnus,FOberhettinger,andFTricomi,Higher Transcendental Functions, vol 2, Based on notes left by Harry Bateman, McGraw-Hill, New Yor, 953 MR 5:49i

10 058 V V KARACHIK [5] E D Rainville, Special Functions, The Macmillan Company, New Yor, 960 MR 2:6447 [6] V S Vladimirov, Uravneniya matematichesoj fizii, Fizio-Matematichesaya Literatura, Mosva, 2000 Russian) 48, Behterev St, Apt 6, 70005, Tashent, Uzbeistan address: arachi@uwedfreenetuz Current address: Institute of Cybernetics, Uzbe Academy of Sciences, 34, F Hodzhaev St, 70025, Tashent, Uzbeistan address: arachi@ttuz