Populate a column based on previous value and row Pyspark

Within a window you cannot access results of a column that you are currently about to calculate. This would force Spark to do the calculations sequentially and should be avoided. Another approach is to transform the recursive calculation c_n = func(c_(n-1)) into a formula that only uses the (constant) values of a, b and the first value of c:

All input values for this formula can be collected with a window and the formula itself is implemented as udf:

from pyspark.sql import functions as F
from pyspark.sql import types as T
from pyspark.sql import Window

df = ...
w=Window.partitionBy('group').orderBy('date')
df1 = df.withColumn("la", F.collect_list("a").over(w)) \
  .withColumn("lb", F.collect_list("b").over(w)) \
  .withColumn("c0", F.first("c").over(w))

import numpy as np

def calc_c(c0, a, b):
  if c0 is None:
    return 0.0
  if len(a) == 1:
    return float(c0)
  e1 = c0 * np.prod(b[:-1])
  e2 = 0.0
  for i,an in enumerate(a[:-1]):
    e2 = e2 + an * np.prod(b[i+1:-1])
  return float(e1 + e2)


calc_c_udf= F.udf(calc_c, T.DoubleType())

df1.withColumn("result", calc_c_udf("c0", "la", "lb")) \
  .show()

Output:

+-----+-------+---+---+----+---------+---------+---+------+
|group|   date|  a|  b|   c|       la|       lb| c0|result|
+-----+-------+---+---+----+---------+---------+---+------+
|    a|2018-01|  2|  3|  10|      [2]|      [3]| 10|  10.0|
|    a|2018-02|  4|  5|null|   [2, 4]|   [3, 5]| 10|  32.0|
|    a|2018-03|  2|  1|null|[2, 4, 2]|[3, 5, 1]| 10| 164.0|
+-----+-------+---+---+----+---------+---------+---+------+