A NOTE ON JAIN BASIS FUNCTIONS
arXiv:1612.09385v1 [math.CA] 30 Dec 2016
G. C. Greubel Newport News, VA, United States
[email protected] Abstract. In the present article the moments associated with the Jain basis functions are developed to order ten. The moments are shown to be polynomials in one variable with polynomial coefficients. The polynomial coefficients are investigated and shown to be linked to sequences found in the On-line Encyclopedia of Integer Sequences. Keywords. Bell polynomials, Eulerian numbers of the first and second kind, Jain basis, Oeis, Stirling numbers of the second kind. Oeis Sequences. A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A000914, A000915, A001287, A001303, A008292, A008517, A053567. 1. Introduction G. C. Jain, [1], introduced the operators ∞ X k (β) β f Pn (f, x) = Ln,k (x), n
x ∈ [0, ∞),
(1)
k=0
where 0 ≤ β < 1 and the basis functions are defined as (β)
Ln,k (x) =
nx (nx + kβ)k−1 −(nx+kβ) e , k!
(2)
P (β) and has the property ∞ k=0 Ln,k (x) = 1 are defined for 0 ≤ β < 1 and reduce to the SzaszMirakyan operators when β = 0. Jain introduced the method to obtain the moments to be used with these operators. In order to define the moments of the generalized Szasz-Mirakyan operators the series S(r, α, β) is to be defined as S(r, α, β) =
∞ X 1 (α + βk)k+r−1 e−(α+βk) , k!
r = 0, 1, 2, 3, · · ·
(3)
k=0
where α S(0, α, β) = 1 and S(r, α, β) =
∞ X
β k (α + βk) S(r − 1, α + βk, β).
(4)
k=0
When f (t) = tm in (1), where Pnβ (f = tm ; x) = Bnβ (tm , x), the operators become the desired moments used in the referenced papers. The first couple of moments are obtained by following: Bnβ (1, x) = 1, ∞ X (nx + kβ)k−1 −(nx+kβ) k β e = x S(1, nx + β, β) Bn (t; x) = nx k! n k=1
1
2
MOMENT ESTIMATES
and Bnβ (t2 ; x)
= nx
∞ X (nx + kβ)k−1
k!
k=0
−(nx+kβ)
e
2 x k = [S(2, nx + 2β, β) + S(1, nx + β, β)] . n n
Jain presented the first three moments which, for most cases, is enough to establish convergence of the operators and be used to identify relations dependent upon these moments. In recent works, Gupta & Greubel [2, 3], Gupta & Malik [4], and others have extended the list of moments generaly used to fifth order. Extending the list of moments helps establish higher orders of convergence and can lead to new identities dependent upon these moments. In this article the list of moments is extended to order ten. The coefficients, of the variable β will be given and examined. The coefficients of these polynomials will be linked to sequences given in the On-line Encyclopedia of Integer Sequences (Oeis) [5].
2. Developement of S(r, α, β) First define the Eulerian polynomials, An (x), by ∞ X
k n xk =
k=0
Pn
x An (x) , (1 − x)n+1
(5)
A(n, m) xm
where An (x) = j=0 and A(n, m) are the Eulerian numbers. Eulerian numbers are listed as sequence A008292 in Oeis and the first few polynomials are given by A0 (x) = 1 and A1 (x) = 1 A2 (x) = 1 + x A3 (x) = 1 + 4 x + x2
(6) 2
A4 (x) = 1 + 11 x + 11 x + x
3
A5 (x) = 1 + 26 x + 66 x2 + 26 x3 + x4 . Beginning with (3, 4) it can be seen that S(1, α, β) =
∞ X
β k (α + βk) S(0, α + βk, β) =
k=0
∞ X
βk =
k=0
1 = p. 1−β
(7)
The next couple are given by the following: S(2, α, β) =
∞ X
β k (α + βk) S(1, α + βk, β) = p
k=0
∞ X
β k (α + βk)
k=0
= p (α p A0 (β) + β p A1 (β)) = p (α p + β 2 p2 ),
S(3, α, β) =
∞ X
2 2
β k (α + βk) S(2, α + βk, β) = p2
k=0 2 2
∞ X
(8)
β k [(α + βk)2 + p3 (α + βk) β 2 ]
k=0
2
3
= p (α p A0 (β) + 2α β A1 (β) + p β A2 (β)) + p3 β 2 (α p β A0 (β) + β 2 p2 A1 (β)) = α2 p3 + 3 α β 2 p4 + β 3 (1 + 2β) p5 .
3
(9)
MOMENT ESTIMATES
3
Repeating the process a list can be generated for the polynomials S(r, α, β). The polynomials are given by: S(0, α, β) =
1 α
1 S(1, α, β) = p = 1−β "
S(r, α, β) = pr αr−1 +
r−1 X
#
θkr (β) αr−k−1 β k+1 pk ,
k=1
r ≥ 2.
(10)
It is quickly determined that θ1r (β) ∈ {0, 1, 3, 6, 10, 15, 21, 28, 36, 45, · · · }r≥1 which is sequence A000217 and is the binomial coefficients r2 , ie. θ1r (β) = 2r . r θ2r (β) θ3r (β) θ4r (β) 2 0 0 0 3 1 + 2β 0 0 4 + 11 β 1 + 8 β + 6 β2 0 4 10 + 35 β 5 + 50 β + 50 β 2 1 + 22 β + 58 β 2 + 24 β 3 5 6 20 + 85 β 15 + 180 β + 225 β 2 6 + 157 β + 508 β 2 + 274 β 3 2 35 + 490 β + 735 β 21 + 637 β + 2443 β 2 + 1624 β 3 7 35 + 175 β 2 8 56 + 322 β 70 + 1120 β + 1960 β 56 + 1932 β + 8568 β 2 + 6769 β 3 2 126 + 4872 β + 24528 β 2 + 22449 β 3 9 84 + 546 β 126 + 2268 β + 4536 β 10 120 + 870 β 210 + 4200 β + 9450 β 2 252 + 10794 β + 60816 β 2 + 63273 β 3 θ56 (β) = 1 + 52β + 328β 2 + 444β 3 + 120β 4 θ57 (β) = 7 + 420β + 3108β 2 + 5096β 3 + 1764β 4 θ58 (β) = 28 + 1904β + 16170β 2 + 31136β 3 + 13132β 4 θ59 (β) = 84 + 6384β + 61194β 2 + 135324β 3 + 67284β 4 θ510 (β) = 210 + 17640β + 188370β 2 + 470400β 3 + 269325β 4 θ67 (β) = 1 + 114β + 1452β 2 + 4400β 3 + 3708β 4 + 720β 5 θ68 (β) = 8 + 1031β + 14976β 2 + 52756β 3 + 53296β 4 + 13068β 5 θ69 (β) = 36 + 5175β + 84430β 2 + 339000β 3 + 399180β 4 + 118124β 5 θ610 (β) = 120 + 19035β + 344620β 2 + 1553430β 3 + 2088840β 4 + 723680β 5 θ78 (β) = 1 + 240β + 5610β 2 + 32120β 3 + 58140β 3 + 33984β 5 + 5040β 6 θ79 (β) = 9 + 2406β + 62910β 2 + 407880β 3 + 852180β 4 + 592056β 5 + 109584β 6 θ710 (β) = 45 + 13260β + 383475β 2 + 2777040β 3 + 6577080β 4 + 5292600β 5 + 1172700β 6 θ89 (β) = 1 + 494β + 19950β 2 + 195800β 3 + 644020β 4 + 785304β 5 + 341136β 6 + 40320β 7
4
MOMENT ESTIMATES
θ810 (β) = 10 + 5441β + 242364β 2 + 2645150β 3 + 9799480β 4 + 13711620β 5 + 7028784β 6 + 1026576β 7 θ910 (β) = 1 + 1004β + 67260β 2 + 1062500β 3 + 5765500β 4 + 12440064β 5 + 11026296β 6 + 3733920β 7 + 362880β 8 What is first noticed is that the β 0 coefficients of the θkr (β) polynomials have the form These correspond to sequences A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287, respectively for k = 2, · · · , 10. By comparing the coefficients of r (β) = r + s(r + 1, r − 1) β = and A000914 it is determined that θ θ2r(β) to sequences A000292 2 3 r 1 (3r − 1)β , where s(n, m) are Stirling numbers of the first kind. Comparing the 1 + 4 3 coefficients of θ3r (β) to A000332 and A001303 leads to r r r r(r − 1) 2 r 2 θ3 (β) = + 2r β + s(r + 2, r − 1) β = 1 + 2rβ + β . (11) 4 4 4 2 r k .
Similarly comparing to A000389 and A000915 θ4r (β) has the form r r θ4 (β) = + α41 β + α42 β 2 + s(n + 3, r − 1) β 3 5 1 1 r 1 2 2 3 2 3 (15r − 30r + 5r + 2) β . (12) = 1 + (25r + 7)β + (15r − 5r − 2) β + 6 6 48 5 Comparing A000579 and A053567 then 1 r 1 r θ5 (β) = 1 + (8r + 4) β + (35r 2 + 9r − 2) β 2 + r(5r 2 − 5r − 2) β 3 4 2 6 1 + r(r − 1)(3r 2 − 7r − 2) β 3 . 16
(13)
Further exact forms of the polynomials θkr (β) can be obtained by expanding the list of polynomials. This is currently beyond the scope of this work.
3. Developement of S(r, nx + rβ, β) Since S(r, α, β) has been desired it can be quickly determined that a similar form will be obtained when α → y + rβ, where y = nx. The listing of polynomials will be given in the following list. The polynomials are given by: S(0, y, β) =
1 y
1 S(1, y + β, β) = p = 1−β "
S(r, y + rβ, β) = p The list of polynomials in β is:
r
y
r−1
+
r−1 X k=1
φrk (β) y r−k−1 β k
p
k
#
,
r ≥ 2.
(14)
MOMENT ESTIMATES
r 2 3 4 5 6 7 8 9 10
φr1 (β) φr2 (β) 1 (2 - β) 0 3 (2 - β) 9 - 8 β + 2 β2 6 (2 - β) 48 - 44 β + 11 β 2 64 - 79 10 (2 - β) 150 - 140 β + 35 β 2 500 - 645 15 (2 - β) 360 - 340 β + 85 β 2 2160 - 2865 2 21 (2 - β) 735 - 700 β + 175 β 6860 - 9275 28 (2 - β) 1344 - 1288 β + 322 β 2 17920 - 24570 36 (2 - β) 2268 - 2184 β + 546 β 2 40824 - 56574 45 (2 - β) 3600 - 3480 β + 870 β 2 84000 - 117390
5
φr3 (β) 0 0 β + 36 β 2 - 6 β 3 β + 300 β 2 - 50 β 3 β + 1350 β 2 - 225 β 3 β + 4410 β 2 - 735 β 3 β + 11760 β 2 - 1960 β 3 β + 27216 β 2 - 4536 β 3 β + 56112 β 2 - 13272 β 3
φ54 (β) = 625 − 974β + 622β 2 − 192β 3 + 24β 4 φ64 (β) = 6480 − 10614β + 6997β 2 − 2192β 3 + 274β 4 φ74 (β) = 36015 − 60984β + 41062β 2 − 12992β 3 + 1624β 4 φ84 (β) = 143360 − 248584β + 169932β 2 − 54152β 3 + 6769β 4 φ94 (β) = 459270 − 810684β + 560532β 2 − 179592β 3 + 22449β 4 2 3 4 φ10 4 (β) = 1260000 − 2255148β + 1537914β − 700224β + 292593β
φ65 (β) = 7776 − 14543β + 11758β 2 − 5126β 3 + 1200β 4 − 120β 5 φ75 (β) = 100842 − 198639β + 166824β 2 − 74508β 3 + 17640β 3 − 1764β 5 φ85 (β) = 688128 − 1405348β + 1211728β 2 − 550326β 3 + 131320β 4 − 13132β 5 φ95 (β) = 3306744 − 6935460β + 6096720β 2 − 2803290β 3 + 672840β 4 − 67284β 5 2 3 4 5 φ10 5 (β) = 12600000 − 26972190β + 24064740β − 11170530β + 2693250β − 269325β
φ76 (β) = 117649 − 255828β + 248250β 2 − 137512β 3 + 45756β 3 − 8640β 5 + 720β 6 φ86 (β) = 1835008 − 4203576β + 4251207β 2 − 2426848β 3 + 823092β 4 − 156816β 5 + 13068β 6 φ96 (β) = 14880348 − 35392104β + 36865845β 2 − 21501260β 3 + 7392480β 4 − 1417488β 5 + 118124β 6 2 3 4 φ10 6 (β) = 84000000 − 205539480β + 218906235β − 129746180β + 45069630β
− 8684160β 5 + 723680β 6 φ87 (β) = 2097152 − 5187775β + 5846760β 2 − 3892430β 3 + 1651480β 4 − 445572β 5 + 70560β 6 − 5040β 7 θ79 (β) = 38263752 − 99640521β + 117218964β 2 − 80719830β 3 + 35097840β 4 − 9617772β 5 + 1534176β 6 − 109584β 7
6
MOMENT ESTIMATES
2 3 4 φ10 7 (β) = 360000000 − 973850355β + 1182292710β − 834468975β + 369371340β
− 102373620β 5 + 16417800β 6 − 1172700β 7 φ98 (β) = 43046721 − 119214746β + 152606870β 2 − 118016760β 3 + 60289700β 4 − 20808776β 5 + 4728816β 6 − 645120β 7 + 40320β 8 2 3 4 φ10 8 (β) = 900000000 − 2620523090β + 3503248241β − 2808432336β + 1475733950β
− 519839720β 5 + 119671020β 6 − 16425216β 7 + 1026576β 8 2 3 4 φ10 9 (β) = 1000000000 − 3062575399β + 3793195314β − 4159084510β + 16269598360β
− 46433648700β 5 + 68381809464β 6 − 54285272064β 7 + 22150947840β 8 − 3620510880β 9 It is quickly determined that φr1 (β) = 2r (2 − β). The next few exact forms are given by: 1 r r (12r − 4(3r − 1)β + (3r − 1)β 2 ) (15) φ2 (β) = 4 3 1 r r φ3 (β) = (8r 2 − 2(6r 2 − 4r − 1)β + 6r(r − 1)β 2 − r(r − 1)β 3 ) (16) 2 4 1 r (240r 3 − 48(10r 3 − 10r 2 − 5r − 1)β + 8(45r 3 − 75r 2 − 5r + 7)β 2 φr4 (β) = 48 5
− 8(15r 3 − 30r 2 + 5r + 2)β 3 + (15r 3 − 30r 2 + 5r + 2)β 4 ) (17) 1 r φr5 (β) = (96r 4 − 16(15r 4 − 20r 3 − 15r 2 − 6r − 1)β + 16(15r 4 − 35r 3 − 5r 2 + 9r + 4)β 2 16 6 + 4(30r 4 − 90r 3 + 25r 2 + 27r + 2)β 3 + 10r(r − 1)(3r 2 − 7r − 2)β 4 + r(r − 1)(3r 2 − 7r − 2)β 5 ).
(18)
Equations (14 - 18) will be essential in providing the exact forms of the first few Jain moments.
4. Jain Moments Bnβ (tm , x) In the introduction section the first couple of moments were shown how they connect to S(r, nx + rβ, β). The general form is given by, where y = nx, Bnβ (tm , x) =
m y X S(m, r) S(r, y + rβ, β), nm r=1
where S(m, r) are the Stirling numbers of the second kind. Up to m = 5 the moments are Bnβ (t0 , x) = 1 yp Bnβ (t1 , x) = n 2 yp Bnβ (t2 , x) = 2 (y + p) n 3 yp Bnβ (t3 , x) = 3 (y 2 + 3yp + (1 + 2β)p2 ) n
(19)
MOMENT ESTIMATES
7
yp4 3 (y + 6y 2 p + (7 + 8β) yp2 + (1 + 8β + 6β 2 )p3 ) n4 yp5 Bnβ (t5 , x) = 5 (y 4 + 10y 3 p + 5(5 + 4β) y 2 p2 + 15(1 + 4β + 2β 2 ) yp3 n + (1 + 22β + 58β 2 + 24β 3 )p4 ). Bnβ (t4 , x) =
From this list it can be seen that the general form is " # m−2 m X m y p y m−2 p + σkm (β) y m−k−1 pk + Bm−1 (β) pm−1 , Bnβ (tm , x) = m y m−1 + n 2
(20)
k=2
where Bn (x) are the Eulerian polynomials of the second kind. The coefficients of the Eulerian polynomials of the second kind are given by sequence A008517. The first few σkm (β) polynomials are given by 1 m m σ2 (β) = (3m − 5 + 8β) (21) 4 3 1 m m ((m − 2)(m − 3) + 8(m − 2)β + 12β 2 ) (22) σ3 (β) = 2 4 1 m σ4m (β) = ((15m3 − 150m2 + 485m − 502) + 16(15m2 − 95m + 116)β 48 5 + 16(65m − 151)β 2 + 1152β 3 ) 1 m σ5m (β) = ((3m4 − 50m3 + 305m2 − 802m + 760) + 16(5m3 − 55m2 16 6
+ 196m − 224)β + 8(85m2 − 537m + 818)β 2 + 192(11m − 29)β 3 + 120β 4 ).
(23)
(24)
With the exact forms of (20 - 24) many of the polynomial coefficients in the moments can be quickly determined. The remaining terms not presented here, in exact form, required calculations beyond the scope of this work. While most of the articles written involving the use of Jain moments use up to m = 5 the above work yields all necessary terms. The listing of moments up to m = 10 is: Bnβ (t0 , x) = 1 yp Bnβ (t1 , x) = n 2 yp Bnβ (t2 , x) = 2 (y + p) n 3 yp Bnβ (t3 , x) = 3 (y 2 + 3yp + (1 + 2β)p2 ) n 4 yp Bnβ (t4 , x) = 4 (y 3 + 6y 2 p + (7 + 8β) yp2 + (1 + 8β + 6β 2 )p3 ) n 5 yp Bnβ (t5 , x) = 5 (y 4 + 10y 3 p + 5(5 + 4β) y 2 p2 + 15(1 + 4β + 2β 2 ) yp3 n + (1 + 22β + 58β 2 + 24β 3 )p4 )
8
MOMENT ESTIMATES
Bnβ (t6 , x) =
Bnβ (t7 , x) =
yp6 5 (y + 15y 4 p + 5(13 + 8β)y 3 p2 + 30(3 + 8β + 3β 2 )y 2 p3 + (31 + 292β n6 + 478β 2 + 144β 3 )yp4 + (1 + 52β + 328β 2 + 444β 3 + 120β 4 )p5 ) yp7 6 (y + 21y 5 p + 70(2 + β)y 4 p2 + 70(5 + 10β + 3β 2 )y 3 p3 + 7(43 + 256β n7 + 304β 2 + 72β 3 )y 2 p4 + 21(3 + 56β + 204β 2 + 192β 3 + 40β 4 )yp5 + (1 + 114β + 1452β 2 + 4400β 3 + 3708β 4 + 720β 5 )p6 )
Bnβ (t8 , x) =
yp8 7 (y + 28y 6 p + 14(19 + 8β)y 5 p2 + 210(5 + 8β + 2β 2 )y 4 p3 + 21(81 + 352β n8 + 328β 2 + 64β 3 )y 3 p4 + 42(23 + 256β + 654β 2 + 472β 3 + 80β 4 )y 2 p5 + (127 + 4272β + 29172β 2 + 58832β 3 + 36972β 4 + 5760β 5 )yp6 + (1 + 240β + 5610β 2 + 32120β 3 + 58140β 4 + 33984β 5 + 5040β 6 )p7 )
Bnβ (t9 , x) =
yp9 8 (y + 36y 7 p + 42(11 + 4β)y 6 p2 + 126(21 + 28β + 6β 2 )y 5 p3 + 21(331 n9 + 1132β + 868β 2 + 144β 3 )y 4 p4 + 210(37 + 292β + 574β 2 + 336β 3 + 48β 4 )y 3 p5 + 5(605 + 11388β + 52656β 2 + 79504β 3 + 39852β 4 + 5184β 5 )y 2 p6 + 15(17 + 972β + 11286β 2 + 41200β 3 + 53964β 4 + 24672β 5 + 3024β 6 )yp7 + (1 + 494β + 19950β 2 + 195800β 3 + 644020β 4 + 785304β 5 + 341136β 6 + 40320β 7 )p8 )
Bnβ (t10 , x) =
yp10 9 (y + 45y 8 p + 30(25 + 8β)y 7 p2 + 420(14 + 16β + 3β 2 )y 6 p3 + 21(1087 n10 + 3064β + 1996β 2 + 288β 3 )y 5 p4 + 315(135 + 824β + 1316β 2 + 648β 3 + 80β 4 )y 4 p5 + 5(6821 + 88248β + 307680β 2 + 371152β 3 + 154764β 4 + 17280β 5 )y 3 p6 + 30(311 + 9288β + 70401β 2 + 187864β 3 + 192732β 4 + 72384β 5 + 7560β 6 )y 2 p7 + (511 + 47804β + 886650β 2 + 5317280β 3 + 12298780β 4 + 11470464β 5 + 4034736β 6 + 403200β 7 )yp8 + (1 + 1004β + 67260β 2 + 1062500β 3 + 5765500β 4 + 12440064β 5 + 11026296β 6 + 3733920β 7 + 362880β 8 )p9 )
References [1] G. C. Jain, Approximation of functions by a new class of linear operators, J. Austral. Math. Soc. 13(3), 1972, 271-276 [2] V. Gupta and G. C. Greubel, Moment Estimations of new Sz´ asz-Mirakyan-Durrmeyer operators, Appl. Math. Comput. 271 (2015), 540-547. [3] V. Gupta and G. C. Greubel, A Note of modified Phillips operators, arXiv:1604.008847 [4] V. Gupta and N. Malik, Direct Estimations of new generalized Baskakov-Sz´ asz operators, Publications De L’Institut Mathe´ematique, Nouvelle s´erie, tome 99(113), 2016, 265-279 [5] On-line Encyclopedia of Integer Sequences, oeis.org
