One secondary source has Polly married to Samuel SMITH; another secondary source has her married to both Asher DAVIS and Samuel SMITH.  I have not found a marriage record with a Samuel SMITH, either before or after her marriage to Asher.  Given her youth when she married Asher, if she did marry Samuel, it was as her second marriage and after 1850, so in any case, Samuel SMITH was not the father of any children with her (she was too old to bear any children after 1850).

— born in Trigg Co., KY:
  5.  (Son B) DAVIS, b. 1820
  6.  (Son C) DAVIS, b. 1820-25
  7.  Messiniah DAVIS, b. 1824/5 
  8.  (Son D) DAVIS, b. 1825-30
  9.  John DAVIS, b. 1826/7

— born in Trigg or Christian Co., KY
10.  Andrew DAVIS, b. 1830/1

— born in Christian Co., KY:
11.  (Daughter D) DAVIS, b. 1835-40

1.  Jordan R. Dodd, ed.  1993.  Kentucky Marriages:  Early to 1850.  Liahona Research, Orem, UT (Broderbund CD-229):

Davis, Asher C.

Rascoe, Polly

Dec 31, 1811

Christian Co.

2a.  Census Index:  U.S. Selected Counties, 1820 (Broderbund CD-314):

1820

Asher, Davis

KY

Trigg Co.

p. 4

2b.  Archives of the DAVIS-L mailing list (online at RootsWeb.com).

Trigg County Census Records of 1820 by Don Simmons, Published 1987. Page 4 Davis, Asher C. Males: [1] 0-10, [1] 16-26 Females: [3] 0-10, [1] 16-26

2c.  1820 Census Index and Digital Images (online at Ancestry.com; Image #4 of 19; extracted by Diana Gale Matthiesen):

1820

KY

Trigg Co.

p. ?

Asher C. Davis

100100-30100-0-100

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	9 or under	1810-1820	= Son A
1 male	19-25*	1794-1801	= Asher (b. 1792/3)
3 females	9 or under	1810-1820	= Daughter C = Daughter B = Daughter A
1 female	16-25	1794-1804	= Mary (b. 1796/7)
*In the 1820 Census, the third column is age class 16-18 and the fourth column is age class 16-25; therefore, any individual in column three is duplicated in column four.  By subtracting the number in column three from the number in column four, you can create an age class "19-25."

Next door to John DAVIS, who is next door to John DAVIS, Sr.

3.  Archives of the DAVIS-L mailing list (online at RootsWeb.com).

Trigg County Census of 1830 by Don Simmons, Published 1987. Page 16 Davis, Asher D.  Males:212001  Females:012101

The "D." must be an error.  These date indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
2 males	4 or under	1825-1830	= Andrew (b. 1830/1) or Son D  = John (b. 1827/8)
1 male	5-9	1820-1825	= Son C
2 males	10-14	1815-1820	= Son B (b. 1820) = Son A
1 male	30-39	1790-1800	= Asher (b. 1792/3)
1 female	5-9	1820-1825	= Messiniah (b. 1823/4)
2 females	10-14	1815-1820	= Daughter C = Daughter B
1 female	15-19	1810-1815	= Daughter A
1 female	30-39	1790-1800	= Mary (b. 1796/7)

4a.  Census Index:  U.S. Selected Counties, 1840 (Broderbund CD-316):

1840

Davis, Aslan C.

KY

Christian Co.

p. 190

4b.  Christian County Genealogical Society (Hopkinsville, KY).  1995.  Christian County, KY, Census Records, 1820-1830-1840.  Simmons Historical Publ., Melber, KY.  On p. 58 (of the book, not the census):

1840

Davis

Asher C.

012 110 100 - 100 110 100

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	5-9	1830-1835	= Andrew (b. 1830/1) or Son D
2 males	10-14	1825-1830	= Andrew (b. 1830/1) or Son D = Son C or John (b. 1827/8)
1 male	15-19	1820-1825	= Son B or C
1 male	20-29	1810-1820	= Son A or B
1 male	40-49	1790-1800	= Asher (b. 1792/3)
1 female	4 or under	1835-1840	= Daughter D
1 female	15-19	1820-1825	= Messiniah (b. 1823/4)
1 female	20-29	1810-1820	= Daughter B or C
1 female	40-49	1790-1800	= Mary (b. 1796/7)

5.  1850 Census Index/Microfilm (online at GenealogyLibrary.com):  Ballard Co., KY, p. 291B, 578/578, enumerated 28 Aug 1850 (extracted by Diana Gale Matthiesen):

Asher Davis	57	M	Farmer		NJ
Mary    "	53	F			Ky
John    "	22	M	Farmer		Ky
Andrew  "	19	M	Farmer		Ky

His birthplace is definitely written NJ, not NC.  Of course, it should be NC, so is this census-taker error?

6.  Willard Rouse Jillson.  1925.  The Kentucky Land Grants. Filson Club Publ., Louisville, KY (online at Ancestry.com; Broderbund CD-650):

Vol. 1, Part 1, Chap. IV, Grants South of Green River (1797-1866)
Pg	Grantee	Acres	Book	Pg	Date	County	Watercourse
296	Davis, Asher	300	20	179	30 Sep 1807	Christian	Potts Cr

Potts Creek is a tributary of the Little River, southeast of Cadiz, which is, today, in Trigg Co., KY.

7.  Trigg County OLIVER and DAVIS Deed Abstracts [link died] (abstracted by Willis Patrick Oliver and online at the Caldwell County KYGenWeb site):

Trigg Co. Deed Book D, p. 141, 20 Oct. 1830. Asher C. Davis sells to Ebinezer Boyd, both of Trigg for $400 160 acres on Saline Creek in Trigg. /s/ Asher Davis. Wit: Rich'd Dawson and L. Novell. Mary Davis, wife of Asher consented. Entered 21 Mar. 1831.

8.  Stewart County [Tennessee] Selected Will Book Extracts 1812-1916 [link died].

Volume 2... 178 settlement of William Rascoe by admr. Dudley Williams, Rachel Rascoe; legatees: Jesse, Latun, William Rasco, Jessee Fort, Dudley Williams, Asher Davis, John Rasco; May term, 1813.

9.  NARA.  Index to the Compiled Military Service Records for the Volunteer Soldiers Who Served During the War of 1812, M603, 234 Rolls. National Archives and Records Administration, Washington DC (digitized by Direct Data Capture, Provo, UT; online at Ancestry.com):

Surname	Given Name	Middle Initial	Company	Unit	Rank- Induction	Rank- Discharge	Misc	Roll- Box	Roll- Exct
DAVIS	ASHER		Capt. Williams' Co. Mounted Riflemen, Kentucky Volunteers.		Private	Private		53	602

10.  William Henry Perrin, ed.  1844.  County of Trigg Kentucky:  Historical and Biographical.  F.A. Battey Publ. Co., Chicago:

p. 23-24	The third term of the Cirbuit Court was held in the month of November, 1820... Indictments were returned against... Asher C. Davis for assault and battery...
p. 33	Early Constables... A.C. Davis...

11.  LDS.  Family Search: Internet Genealogy Service: Pedigree Resource File (online at FamilySearch.org).

12.  WorldConnect (online at RootsWeb.com).

13.  Broderbund.  World Family Trees.  Vol. 41, Pedigree #1159.