SQL recursion and cte - dates timeseries
For this you need to change the anchor set of the recursive CTE, so each row starts with the dates from the other table before starting the recursive portion.

DECLARE @AmountDays INT = 10

;WITH GeneratedDays AS
(
    -- Anchor
    SELECT
        DateSeriesIdentifier = T.DateSeriesIdentifier,
        Date = T.Date, -- Assuming it's DATE data type
        RecursionLevel = 1
    FROM
        OtherTable AS T

    UNION ALL

    -- Recursion
    SELECT
        DateSeriesIdentifier = G.DateSeriesIdentifier,
        Date = DATEADD(DAY, 1, G.Date),
        RecursionLevel = G.RecursionLevel + 1
    FROM
        GeneratedDays AS G
    WHERE
        G.RecursionLevel + 1 &lt;= @AmountDays
)
SELECT
    G.DateSeriesIdentifier,
    G.Date,
    G.RecursionLevel
FROM
    GeneratedDays AS G
ORDER BY
    G.DateSeriesIdentifier,
    G.RecursionLevel


This is a result from a set I used to test which had dates 2016-02-04, 2018-01-19 and 2018-08-30:

DateSeriesIdentifier    Date            RecursionLevel
21                      2016-02-04      1
21                      2016-02-05      2
21                      2016-02-06      3
21                      2016-02-07      4
21                      2016-02-08      5
21                      2016-02-09      6
21                      2016-02-10      7
21                      2016-02-11      8
21                      2016-02-12      9
21                      2016-02-13      10
620646                  2018-01-19      1
620646                  2018-01-20      2
620646                  2018-01-21      3
620646                  2018-01-22      4
620646                  2018-01-23      5
620646                  2018-01-24      6
620646                  2018-01-25      7
620646                  2018-01-26      8
620646                  2018-01-27      9
620646                  2018-01-28      10
639701                  2018-08-30      1
639701                  2018-08-31      2
639701                  2018-09-01      3
639701                  2018-09-02      4
639701                  2018-09-03      5
639701                  2018-09-04      6
639701                  2018-09-05      7
639701                  2018-09-06      8
639701                  2018-09-07      9
639701                  2018-09-08      10